TRIANGLE CONSTRUCTION A, h_b+a, h_c+a

To construct triangle ABC if are given A, h_b + a, h_c + a, where h_b,h_c are the altitudes from B,C, resp.

Solution 1

We have:

2Rh_b = ac
2Rh_c = ab

==>

h_b + a = (ac/2R) + a = (a/2R).(2R + c)

h_c + a = (ab/2R) + a = (a/2R).(2R + b)

We have that angle A is known ==> a/2R is known

[Geometric Proof:

Let O be the circumcenter and M the midpoint of BC.
The triangle BOC has known angles since A is known ==>

MC / OC is known ==> (a/2)/R = a/2R is known].

So the problem is equivalent to construct triangle if are given A, 2R + b, 2R + c.

Analysis:

Let ABC be the triangle in question, AOD diameter of the circumcircle and B',C' points on the extensions of AB,AC such that BB' = CC' = AD [= 2R].

A is known

AB' = AB + BB' = c + 2R, known

AC' = AC + CC' = b + 2R, known

==> The triangle AB'C' can be constructed.

We have:

DB is perpendicular to AB' (since AD is diameter) and BB' = AD [=2R]

==> the locus of D is the parabola with focus A and directrix the perpendicular to AB' at B' (see LEMMA).

Similarly:

DC is perpendicular to AC' and CC' = AD ==> the locus of D is the parabola with focus A and directric the perpendicular to AC' at C'.

Therefore D is intersection point of the two loci. B,C are the (other than A) intersections of the circle of diameter AD with the lines AB',AC' resp.

LEMMA:

In triangle ABC, let BC be fixed and D the orthogonal projection of A on BC. If BD = AC then the locus of A is a parabola.

Let A' be the orthogonal projection of A on the perpendicular to BC at B. We have AC = BD and BD = AA' ==> AC = AA' ==> the locus of A is the parabola with focus C and directrix the perpendicular to BC at B.

Solution 2

Let BB' = h_b, CC' = h_c be the altitudes from B,C, resp.

The right triangles C'AC,B'AB are similar and have known angles (since A is known)

==>

CC' / CA = BB' / BA is known ==>

h_c / b = h_b / c = (h_c - h_b) / (b - c) = [(h_c + a) - (h_b + a)] / (b - c)

==> b - c is known.

So the problem is equivalent to construct triangle if are given A, b - c, h_c + a.

Analysis:

Let ABC be the triangle in question with AC > AB.

Let D be the point on AC between A and C such that AD = AB, CE the altitude from C, and Z the intersection of the lines BD and CE.

In the triangle CDZ we have:

CD = AC - AD = b - c, known.

Angles (BDC) = (DAB) + (DBA) = A + (90 - (A/2)) = 90 + (A/2), known

(DCZ) = 90 - (CAE) = 90 - A, known.

Therefore the triangle CDZ can be constructed.

Let H be the point on the extension of CE such that EH = BC.

We have CH = CE + EH = h_c + a, known.

BE is perpendicular to CH and BC = EH ==> (according to LEMMA) the locus of B is the parabola with focus C and directrix the perpendicular to CH at H.

So B is the intersection of the line DZ and the parabola.

Solution 3

Let ABC be the triangle in question and O its circumcenter.

The perpendicular bisector of BC intersects the circumcircle at D,E (as in the figure).

Denote:

CD = DB =: m

EC = EB =: n

AD := d

EA := e

We have:

b + 2R := k1, known (1)

c + 2R := k2, known (2)

m(b + c)= ad (3)(by Ptolemy Theorem in the cyclic quadrilateral ABDC)

The triangle BCD has known angles (DCB = DBC = A/2, CDB = 180 - A)

==> a/m := t is known.

==> b + c = (a/m)d = td

ae + cn = bn (4)(by Ptolemy Theorem in the cyclic quadrilateral ABCE) ==>
e = (b - c)n/a

We have:

b - c = k1 - k2, known

n/a is known since the triangle CEB has known angles (CEB = A, ECB = EBC = 90 - (A/2))

Therefore e = (b - c)n/a is known.

EA^2 = AD^2 - AD^2 (5)(by Pythagorean Theorem in the right triangle ADE)or e^2 = 4R^2 - d^2, known.

Now, from:

(1) and (2) ==> b + c = k1 + k2 - 4R (6)

(6) and (3) ==> d = (k1 + k2 - 4R) / t (7)

(7) and (5) ==> 4R^2 - ((k1 + k2 - 4R)/t)^2 = e^2

== > R is known.

So the problem is equivalent to construct triangle if are given A, b - c, R or A, b - c, a (the solution is left to the reader).

Exercises:

To construct triangle ABC if are given:

1. A, h_b - a, h_c + a

2. A, h_b - a, h_c - a

TRIANGLE CONSTRUCTION A, 2b+a, 2c+a

To construct triangle ABC if are given A, 2b + a = m, 2c + a = n

Analysis

Let ABC be the triangle in question.

We have:

m+n = 2(a+b+c) = 4s ==> the semiperimeter s is known

m-n = 2(b-c) ==> the difference b-c is known

Let E,D the points the a-excircle (Ia) touches AC,BC, resp.
The triangle DAIa has:
ADIa = 90 d., DAIa = A/2, AD = s. Therefore IaD = IaE = r_b is known.

Let M be the midpoint of BC. We have BIaC = 90-(A/2) and ME = (|b-c|)/2.(So the problem is eqivalent to construct triangle if are given:
A, b-c, r_b)

IaM^2 = IaE^2 + ME^2 = (r_b)^2 + ((b-c)/2)^2 ==> the median IaM is known.

In the triangle IaBC we know the angle Ia, the altitude and the median from Ia, therefore the problem is equivalent to construct triangle if are given:

A, h_a, m_a (altitude, median from A, resp.). This construction is left to the reader.

Exercises:

To construct triangle ABC if are given:

1. A, 2b - a = m, 2c + a = n

2. A, 2b - a = m, 2c - a = n

TRIANGLE CONSTRUCTION A, a + b, a + c

To construct triangle ABC if are given A, a + b, a + c

Solution 1.

Analysis:

Let ABC be the triangle in question. Let D, E be two points on the extensions of AC,AB, resp. such that CD = BE = BC = a.

The parallel from A to BC intersects DB at Z.

The triangle AZB is similar to triangle CDB ==>

AZ / AD = CB / CD = 1 ==> AZ = AD (1)

The parallel from Z to AE intersects DE at Q.

The triangles AZQ and CBE are similar ==>

AZ / ZQ = CB / BE = 1 ==> ZQ = AZ (2)

(1) /\ (2) ==> AD = AZ = ZQ (3).

The parallel from Z to EQ intersects AB at H.

We have HE = ZQ (4) (since EHZQ is parallelogram)

(3) /\ (4) ==> HE = AD.

Construction:

I construct the triangle ADE such that AD = b+a, AE = c+a, angle DAE = A.

Let H be on AE such that EH = AD.

The circle (A, AD) intersects the parallel from H to DE at Z.

The line DZ intersects AE at B. The parallel from B to AZ intersects AD at C.

ABC is the required triangle.

The Proof and Investigation are left to the reader.

Reference:
Ioannis Panakis: Solutions of the Exercises of the MATHEMATICS of the 5th Class of Greek Gymnasium, vol I, Athens, Kokotsakis Bookstore, p. 90.

Solution 2.

Analysis:

Let ABC be the triangle in question. Let B', C' be two points on the extensions of AB,AC, resp. such that BB' = CC' = BC = a.

Let D be the intersection of the lines: Parallel from B' to BC and Parallel from C to BB'. The quadrilateral BCDB' is rhombus. The isosceles triangle DCC' has fixed angles: DCC' = A, CDC' = CC'D = (90-A)/2 ("remains similar to itself"). Therefore CD/C'D is fixed, and since CD = B'D ==> B'D/C'D is fixed.

So the point D lies on a known line forming with AC' angle (90-A)/2 and on the Apollonius circle (B'C', B'D/C'D).

The Construction, Proof and Investigation are left to the reader.

Reference:
EUCLID [publ. by the Greek Mathematical Society], December 1982.

Solution 3.

Analysis:

Let ABC be the triangle in question. Let B', C' be two points on the extensions of AB,AC, resp. such that BB' = CC' = BC = a.

Let D be the intersection of BC' and CB' and E the intersection of the parallel from C' to AB' and the parallel from B' to BC'.

The isosceles trianle CC'E has fixed angles:

(CC'E) = 180 - A, (C'CE) = (C'EC) = A/2.

We have:

Angle (CB'E) = (CDC') = (CBD) + (BCD) = C/2 + B/2 = 90 - (A/2) : fixed.

Construction:

I construct the triangle AB'C' such that AB' = c+a, AC' = b+a, angle (B'AC') = A. Let L be an arbitrary point on C'A and M a point on the parallel through C' to AB' such that C'L = C'M. The circle (LM, 90-(A/2)), ie the circle with chord LM and angle 90-(A/2), intersects B'C at N. The parallel through B' to LN intersects AC' at C. The parallel through B' to MN intersects C'M at E. The parallel through C' to B'E intersects AB' at B. The triangle ABC is the required triangle.

Proof (to prove B'B = BC = CC') / Investigation: Left to the reader.

Reference: A. P. Hatzipolakis (1982)

Exercise:
Contruct the triangle if are given:
1. A, b + a, c - a
2. A, b - a, c - a

TRIANGLE RESOLUTION A, a+b-c, R+r

Problem:
To resolve triangle ABC if are given A, a+b-c, R+r
See Hyacinthos Message #19749

Resolution:

We have:

sinA + sinB - sinC = 4sin(A/2)sin(B/2)cos(C/2)

r = 4Rsin(A/2)sin(B/2)sin(C/2)

==>

a+b-c = 2R(sinA+sinB-sinC) = 8Rsin(A/2)sin(B/2)cos(C/2)

R+r = R(1+4sin(A/2)sin(B/2)sin(C/2))

==>

t := (R+r)/(a+b-c) =

= (1+4sin(A/2)sin(B/2)sin(C/2))/8sin(A/2)sin(B/2)cos(C/2)

==>

4sin(A/2)sin(B/2)[sin(C/2) - 2tcos(C/2)] + 1 = 0

and since sin(B/2) = sin(90 - ((C+A)/2)) = cos((C+A)/2) =

= cos(C/2)cos(A/2) - sin(C/2)sin(A/2), ==>

4sin(A/2)[cos(A/2) + 2tsin(A/2)]sin(C/2)cos(C/2) - 4(sin(A/2)^2(sin(C/2))^2 - 8tsin(A/2)cos(A/2)(cos(A/2)^2 + 1 = 0.

We have the system of equations:

fx^2 + gy^2 + hxy + 1 = 0

x^2 + y^2 = 1

where x, y stand for the unknown sin(C/2),cos(C/2) and f,g,h are known coefficients. From these equations we get the equation:

Lx^4 + Mx^2 + N = 0, where L,M,N are known coefficients.

This equation has constructible roots [Read THIS], therefore the problem has a constructible Euclidean solution (ie by ruler and compass)

A CONFIGURATION (Case 3)

See THIS

Let ABC be a triangle and P a point.

The circle (B, BP) intersects AB at Ac1 (between A,B) and Ac2 (on the extension of BA) AND BC at A1c (between B,C) and A2c (on the extension of BC). The circle (C, CP) intersects AC at Ab1 (between A,C) and Ab2 AND BC at A1b (between B,C) and A2b (on the extension of CB).

3. The lines Ab1A1b and Ac1A1c intersect at A3.

Which is the locus of P such that ABC, A3B3C3 are perspective?

Solution
by Francisco Javier García Capitán

The locus looks like Neuberg cubic + three circle arcs:

This is Neuberg Cubic:

Neuberg cubic in dotted line with the locus, showing that they are not exactly the same, but very very close:

The "circle arcs":

The algebraic curve of the eight degree of which the arcs are part, that is they are not really circle arcs:

Equations:

1. The Neuberg-like curve is part of a curve of degree 14 with 20178 terms

2. The equation of the octic (in barycentrics):

64 a^6 b^4 c^8 x^6 y^2 + 256 a^5 b^5 c^8 x^6 y^2 -
128 a^4 b^6 c^8 x^6 y^2 - 256 a^3 b^7 c^8 x^6 y^2 +
64 a^2 b^8 c^8 x^6 y^2 + 256 a^4 b^5 c^9 x^6 y^2 +
512 a^3 b^6 c^9 x^6 y^2 - 256 a^2 b^7 c^9 x^6 y^2 -
128 a^4 b^4 c^10 x^6 y^2 - 256 a^3 b^5 c^10 x^6 y^2 +
384 a^2 b^6 c^10 x^6 y^2 - 256 a^2 b^5 c^11 x^6 y^2 +
64 a^2 b^4 c^12 x^6 y^2 - 16 a^8 b^2 c^8 x^5 y^3 -
112 a^7 b^3 c^8 x^5 y^3 - 48 a^6 b^4 c^8 x^5 y^3 +
1136 a^5 b^5 c^8 x^5 y^3 - 496 a^4 b^6 c^8 x^5 y^3 -
528 a^3 b^7 c^8 x^5 y^3 + 48 a^2 b^8 c^8 x^5 y^3 +
16 a b^9 c^8 x^5 y^3 + 48 a^7 b^2 c^9 x^5 y^3 +
256 a^6 b^3 c^9 x^5 y^3 + 208 a^5 b^4 c^9 x^5 y^3 +
768 a^4 b^5 c^9 x^5 y^3 + 1104 a^3 b^6 c^9 x^5 y^3 -
256 a^2 b^7 c^9 x^5 y^3 - 80 a b^8 c^9 x^5 y^3 -
16 a^6 b^2 c^10 x^5 y^3 + 16 a^5 b^3 c^10 x^5 y^3 +
32 a^4 b^4 c^10 x^5 y^3 - 416 a^3 b^5 c^10 x^5 y^3 +
496 a^2 b^6 c^10 x^5 y^3 + 144 a b^7 c^10 x^5 y^3 -
80 a^5 b^2 c^11 x^5 y^3 - 384 a^4 b^3 c^11 x^5 y^3 -
352 a^3 b^4 c^11 x^5 y^3 - 384 a^2 b^5 c^11 x^5 y^3 -
80 a b^6 c^11 x^5 y^3 + 80 a^4 b^2 c^12 x^5 y^3 +
176 a^3 b^3 c^12 x^5 y^3 + 16 a^2 b^4 c^12 x^5 y^3 -
80 a b^5 c^12 x^5 y^3 + 16 a^3 b^2 c^13 x^5 y^3 +
128 a^2 b^3 c^13 x^5 y^3 + 144 a b^4 c^13 x^5 y^3 -
48 a^2 b^2 c^14 x^5 y^3 - 80 a b^3 c^14 x^5 y^3 +
16 a b^2 c^15 x^5 y^3 + a^10 c^8 x^4 y^4 + 10 a^9 b c^8 x^4 y^4 -
19 a^8 b^2 c^8 x^4 y^4 - 392 a^7 b^3 c^8 x^4 y^4 -
494 a^6 b^4 c^8 x^4 y^4 + 1788 a^5 b^5 c^8 x^4 y^4 -
494 a^4 b^6 c^8 x^4 y^4 - 392 a^3 b^7 c^8 x^4 y^4 -
19 a^2 b^8 c^8 x^4 y^4 + 10 a b^9 c^8 x^4 y^4 + b^10 c^8 x^4 y^4 -
6 a^9 c^9 x^4 y^4 - 54 a^8 b c^9 x^4 y^4 + 8 a^7 b^2 c^9 x^4 y^4 +
872 a^6 b^3 c^9 x^4 y^4 + 716 a^5 b^4 c^9 x^4 y^4 +
716 a^4 b^5 c^9 x^4 y^4 + 872 a^3 b^6 c^9 x^4 y^4 +
8 a^2 b^7 c^9 x^4 y^4 - 54 a b^8 c^9 x^4 y^4 - 6 b^9 c^9 x^4 y^4 +
13 a^8 c^10 x^4 y^4 + 104 a^7 b c^10 x^4 y^4 +
140 a^6 b^2 c^10 x^4 y^4 - 168 a^5 b^3 c^10 x^4 y^4 +
334 a^4 b^4 c^10 x^4 y^4 - 168 a^3 b^5 c^10 x^4 y^4 +
140 a^2 b^6 c^10 x^4 y^4 + 104 a b^7 c^10 x^4 y^4 +
13 b^8 c^10 x^4 y^4 - 8 a^7 c^11 x^4 y^4 - 56 a^6 b c^11 x^4 y^4 -
232 a^5 b^2 c^11 x^4 y^4 - 728 a^4 b^3 c^11 x^4 y^4 -
728 a^3 b^4 c^11 x^4 y^4 - 232 a^2 b^5 c^11 x^4 y^4 -
56 a b^6 c^11 x^4 y^4 - 8 b^7 c^11 x^4 y^4 - 14 a^6 c^12 x^4 y^4 -
84 a^5 b c^12 x^4 y^4 + 46 a^4 b^2 c^12 x^4 y^4 +
360 a^3 b^3 c^12 x^4 y^4 + 46 a^2 b^4 c^12 x^4 y^4 -
84 a b^5 c^12 x^4 y^4 - 14 b^6 c^12 x^4 y^4 + 28 a^5 c^13 x^4 y^4 +
140 a^4 b c^13 x^4 y^4 + 120 a^3 b^2 c^13 x^4 y^4 +
120 a^2 b^3 c^13 x^4 y^4 + 140 a b^4 c^13 x^4 y^4 +
28 b^5 c^13 x^4 y^4 - 14 a^4 c^14 x^4 y^4 - 56 a^3 b c^14 x^4 y^4 -
52 a^2 b^2 c^14 x^4 y^4 - 56 a b^3 c^14 x^4 y^4 -
14 b^4 c^14 x^4 y^4 - 8 a^3 c^15 x^4 y^4 - 24 a^2 b c^15 x^4 y^4 -
24 a b^2 c^15 x^4 y^4 - 8 b^3 c^15 x^4 y^4 + 13 a^2 c^16 x^4 y^4 +
26 a b c^16 x^4 y^4 + 13 b^2 c^16 x^4 y^4 - 6 a c^17 x^4 y^4 -
6 b c^17 x^4 y^4 + c^18 x^4 y^4 + 16 a^9 b c^8 x^3 y^5 +
48 a^8 b^2 c^8 x^3 y^5 - 528 a^7 b^3 c^8 x^3 y^5 -
496 a^6 b^4 c^8 x^3 y^5 + 1136 a^5 b^5 c^8 x^3 y^5 -
48 a^4 b^6 c^8 x^3 y^5 - 112 a^3 b^7 c^8 x^3 y^5 -
16 a^2 b^8 c^8 x^3 y^5 - 80 a^8 b c^9 x^3 y^5 -
256 a^7 b^2 c^9 x^3 y^5 + 1104 a^6 b^3 c^9 x^3 y^5 +
768 a^5 b^4 c^9 x^3 y^5 + 208 a^4 b^5 c^9 x^3 y^5 +
256 a^3 b^6 c^9 x^3 y^5 + 48 a^2 b^7 c^9 x^3 y^5 +
144 a^7 b c^10 x^3 y^5 + 496 a^6 b^2 c^10 x^3 y^5 -
416 a^5 b^3 c^10 x^3 y^5 + 32 a^4 b^4 c^10 x^3 y^5 +
16 a^3 b^5 c^10 x^3 y^5 - 16 a^2 b^6 c^10 x^3 y^5 -
80 a^6 b c^11 x^3 y^5 - 384 a^5 b^2 c^11 x^3 y^5 -
352 a^4 b^3 c^11 x^3 y^5 - 384 a^3 b^4 c^11 x^3 y^5 -
80 a^2 b^5 c^11 x^3 y^5 - 80 a^5 b c^12 x^3 y^5 +
16 a^4 b^2 c^12 x^3 y^5 + 176 a^3 b^3 c^12 x^3 y^5 +
80 a^2 b^4 c^12 x^3 y^5 + 144 a^4 b c^13 x^3 y^5 +
128 a^3 b^2 c^13 x^3 y^5 + 16 a^2 b^3 c^13 x^3 y^5 -
80 a^3 b c^14 x^3 y^5 - 48 a^2 b^2 c^14 x^3 y^5 +
16 a^2 b c^15 x^3 y^5 + 64 a^8 b^2 c^8 x^2 y^6 -
256 a^7 b^3 c^8 x^2 y^6 - 128 a^6 b^4 c^8 x^2 y^6 +
256 a^5 b^5 c^8 x^2 y^6 + 64 a^4 b^6 c^8 x^2 y^6 -
256 a^7 b^2 c^9 x^2 y^6 + 512 a^6 b^3 c^9 x^2 y^6 +
256 a^5 b^4 c^9 x^2 y^6 + 384 a^6 b^2 c^10 x^2 y^6 -
256 a^5 b^3 c^10 x^2 y^6 - 128 a^4 b^4 c^10 x^2 y^6 -
256 a^5 b^2 c^11 x^2 y^6 + 64 a^4 b^2 c^12 x^2 y^6 +
384 a^6 b^6 c^6 x^6 y z + 256 a^5 b^7 c^6 x^6 y z -
512 a^4 b^8 c^6 x^6 y z - 256 a^3 b^9 c^6 x^6 y z +
128 a^2 b^10 c^6 x^6 y z + 256 a^5 b^6 c^7 x^6 y z +
1024 a^4 b^7 c^7 x^6 y z + 256 a^3 b^8 c^7 x^6 y z -
512 a^2 b^9 c^7 x^6 y z - 512 a^4 b^6 c^8 x^6 y z +
256 a^3 b^7 c^8 x^6 y z + 768 a^2 b^8 c^8 x^6 y z -
256 a^3 b^6 c^9 x^6 y z - 512 a^2 b^7 c^9 x^6 y z +
128 a^2 b^6 c^10 x^6 y z + 80 a^8 b^4 c^6 x^5 y^2 z -
176 a^7 b^5 c^6 x^5 y^2 z + 1424 a^6 b^6 c^6 x^5 y^2 z +
400 a^5 b^7 c^6 x^5 y^2 z - 1552 a^4 b^8 c^6 x^5 y^2 z -
272 a^3 b^9 c^6 x^5 y^2 z + 48 a^2 b^10 c^6 x^5 y^2 z +
48 a b^11 c^6 x^5 y^2 z + 240 a^7 b^4 c^7 x^5 y^2 z +
768 a^6 b^5 c^7 x^5 y^2 z + 2480 a^5 b^6 c^7 x^5 y^2 z +
2176 a^4 b^7 c^7 x^5 y^2 z + 80 a^3 b^8 c^7 x^5 y^2 z -
384 a^2 b^9 c^7 x^5 y^2 z - 240 a b^10 c^7 x^5 y^2 z -
80 a^6 b^4 c^8 x^5 y^2 z + 1520 a^5 b^5 c^8 x^5 y^2 z +
352 a^4 b^6 c^8 x^5 y^2 z + 352 a^3 b^7 c^8 x^5 y^2 z +
1008 a^2 b^8 c^8 x^5 y^2 z + 432 a b^9 c^8 x^5 y^2 z -
560 a^5 b^4 c^9 x^5 y^2 z - 1024 a^4 b^5 c^9 x^5 y^2 z +
416 a^3 b^6 c^9 x^5 y^2 z - 1152 a^2 b^7 c^9 x^5 y^2 z -
240 a b^8 c^9 x^5 y^2 z + 48 a^4 b^4 c^10 x^5 y^2 z -
848 a^3 b^5 c^10 x^5 y^2 z + 528 a^2 b^6 c^10 x^5 y^2 z -
240 a b^7 c^10 x^5 y^2 z + 272 a^3 b^4 c^11 x^5 y^2 z +
432 a b^6 c^11 x^5 y^2 z - 48 a^2 b^4 c^12 x^5 y^2 z -
240 a b^5 c^12 x^5 y^2 z + 48 a b^4 c^13 x^5 y^2 z -
12 a^10 b^2 c^6 x^4 y^3 z - 24 a^9 b^3 c^6 x^4 y^3 z -
316 a^8 b^4 c^6 x^4 y^3 z - 208 a^7 b^5 c^6 x^4 y^3 z +
1912 a^6 b^6 c^6 x^4 y^3 z + 256 a^5 b^7 c^6 x^4 y^3 z -
1544 a^4 b^8 c^6 x^4 y^3 z - 48 a^3 b^9 c^6 x^4 y^3 z -
44 a^2 b^10 c^6 x^4 y^3 z + 24 a b^11 c^6 x^4 y^3 z +
4 b^12 c^6 x^4 y^3 z + 8 a^9 b^2 c^7 x^4 y^3 z -
120 a^8 b^3 c^7 x^4 y^3 z + 1072 a^7 b^4 c^7 x^4 y^3 z +
3136 a^6 b^5 c^7 x^4 y^3 z + 4480 a^5 b^6 c^7 x^4 y^3 z +
2128 a^4 b^7 c^7 x^4 y^3 z - 304 a^3 b^8 c^7 x^4 y^3 z -
136 a b^10 c^7 x^4 y^3 z - 24 b^11 c^7 x^4 y^3 z +
68 a^8 b^2 c^8 x^4 y^3 z + 368 a^7 b^3 c^8 x^4 y^3 z +
144 a^6 b^4 c^8 x^4 y^3 z + 5376 a^5 b^5 c^8 x^4 y^3 z +
904 a^4 b^6 c^8 x^4 y^3 z + 624 a^3 b^7 c^8 x^4 y^3 z +
368 a^2 b^8 c^8 x^4 y^3 z + 288 a b^9 c^8 x^4 y^3 z +
52 b^10 c^8 x^4 y^3 z - 80 a^7 b^2 c^9 x^4 y^3 z -
1536 a^5 b^4 c^9 x^4 y^3 z - 1824 a^4 b^5 c^9 x^4 y^3 z +
432 a^3 b^6 c^9 x^4 y^3 z - 576 a^2 b^7 c^9 x^4 y^3 z -
224 a b^8 c^9 x^4 y^3 z - 32 b^9 c^9 x^4 y^3 z -
72 a^6 b^2 c^10 x^4 y^3 z - 512 a^5 b^3 c^10 x^4 y^3 z +
136 a^4 b^4 c^10 x^4 y^3 z - 1232 a^3 b^5 c^10 x^4 y^3 z +
184 a^2 b^6 c^10 x^4 y^3 z - 112 a b^7 c^10 x^4 y^3 z -
56 b^8 c^10 x^4 y^3 z + 128 a^5 b^2 c^11 x^4 y^3 z +
208 a^4 b^3 c^11 x^4 y^3 z + 432 a^3 b^4 c^11 x^4 y^3 z +
128 a^2 b^5 c^11 x^4 y^3 z + 336 a b^6 c^11 x^4 y^3 z +
112 b^7 c^11 x^4 y^3 z - 8 a^4 b^2 c^12 x^4 y^3 z +
144 a^3 b^3 c^12 x^4 y^3 z - 16 a^2 b^4 c^12 x^4 y^3 z -
224 a b^5 c^12 x^4 y^3 z - 56 b^6 c^12 x^4 y^3 z -
48 a^3 b^2 c^13 x^4 y^3 z - 64 a^2 b^3 c^13 x^4 y^3 z +
32 a b^4 c^13 x^4 y^3 z - 32 b^5 c^13 x^4 y^3 z +
20 a^2 b^2 c^14 x^4 y^3 z + 24 a b^3 c^14 x^4 y^3 z +
52 b^4 c^14 x^4 y^3 z - 8 a b^2 c^15 x^4 y^3 z -
24 b^3 c^15 x^4 y^3 z + 4 b^2 c^16 x^4 y^3 z +
4 a^12 c^6 x^3 y^4 z + 24 a^11 b c^6 x^3 y^4 z -
44 a^10 b^2 c^6 x^3 y^4 z - 48 a^9 b^3 c^6 x^3 y^4 z -
1544 a^8 b^4 c^6 x^3 y^4 z + 256 a^7 b^5 c^6 x^3 y^4 z +
1912 a^6 b^6 c^6 x^3 y^4 z - 208 a^5 b^7 c^6 x^3 y^4 z -
316 a^4 b^8 c^6 x^3 y^4 z - 24 a^3 b^9 c^6 x^3 y^4 z -
12 a^2 b^10 c^6 x^3 y^4 z - 24 a^11 c^7 x^3 y^4 z -
136 a^10 b c^7 x^3 y^4 z - 304 a^8 b^3 c^7 x^3 y^4 z +
2128 a^7 b^4 c^7 x^3 y^4 z + 4480 a^6 b^5 c^7 x^3 y^4 z +
3136 a^5 b^6 c^7 x^3 y^4 z + 1072 a^4 b^7 c^7 x^3 y^4 z -
120 a^3 b^8 c^7 x^3 y^4 z + 8 a^2 b^9 c^7 x^3 y^4 z +
52 a^10 c^8 x^3 y^4 z + 288 a^9 b c^8 x^3 y^4 z +
368 a^8 b^2 c^8 x^3 y^4 z + 624 a^7 b^3 c^8 x^3 y^4 z +
904 a^6 b^4 c^8 x^3 y^4 z + 5376 a^5 b^5 c^8 x^3 y^4 z +
144 a^4 b^6 c^8 x^3 y^4 z + 368 a^3 b^7 c^8 x^3 y^4 z +
68 a^2 b^8 c^8 x^3 y^4 z - 32 a^9 c^9 x^3 y^4 z -
224 a^8 b c^9 x^3 y^4 z - 576 a^7 b^2 c^9 x^3 y^4 z +
432 a^6 b^3 c^9 x^3 y^4 z - 1824 a^5 b^4 c^9 x^3 y^4 z -
1536 a^4 b^5 c^9 x^3 y^4 z - 80 a^2 b^7 c^9 x^3 y^4 z -
56 a^8 c^10 x^3 y^4 z - 112 a^7 b c^10 x^3 y^4 z +
184 a^6 b^2 c^10 x^3 y^4 z - 1232 a^5 b^3 c^10 x^3 y^4 z +
136 a^4 b^4 c^10 x^3 y^4 z - 512 a^3 b^5 c^10 x^3 y^4 z -
72 a^2 b^6 c^10 x^3 y^4 z + 112 a^7 c^11 x^3 y^4 z +
336 a^6 b c^11 x^3 y^4 z + 128 a^5 b^2 c^11 x^3 y^4 z +
432 a^4 b^3 c^11 x^3 y^4 z + 208 a^3 b^4 c^11 x^3 y^4 z +
128 a^2 b^5 c^11 x^3 y^4 z - 56 a^6 c^12 x^3 y^4 z -
224 a^5 b c^12 x^3 y^4 z - 16 a^4 b^2 c^12 x^3 y^4 z +
144 a^3 b^3 c^12 x^3 y^4 z - 8 a^2 b^4 c^12 x^3 y^4 z -
32 a^5 c^13 x^3 y^4 z + 32 a^4 b c^13 x^3 y^4 z -
64 a^3 b^2 c^13 x^3 y^4 z - 48 a^2 b^3 c^13 x^3 y^4 z +
52 a^4 c^14 x^3 y^4 z + 24 a^3 b c^14 x^3 y^4 z +
20 a^2 b^2 c^14 x^3 y^4 z - 24 a^3 c^15 x^3 y^4 z -
8 a^2 b c^15 x^3 y^4 z + 4 a^2 c^16 x^3 y^4 z +
48 a^11 b c^6 x^2 y^5 z + 48 a^10 b^2 c^6 x^2 y^5 z -
272 a^9 b^3 c^6 x^2 y^5 z - 1552 a^8 b^4 c^6 x^2 y^5 z +
400 a^7 b^5 c^6 x^2 y^5 z + 1424 a^6 b^6 c^6 x^2 y^5 z -
176 a^5 b^7 c^6 x^2 y^5 z + 80 a^4 b^8 c^6 x^2 y^5 z -
240 a^10 b c^7 x^2 y^5 z - 384 a^9 b^2 c^7 x^2 y^5 z +
80 a^8 b^3 c^7 x^2 y^5 z + 2176 a^7 b^4 c^7 x^2 y^5 z +
2480 a^6 b^5 c^7 x^2 y^5 z + 768 a^5 b^6 c^7 x^2 y^5 z +
240 a^4 b^7 c^7 x^2 y^5 z + 432 a^9 b c^8 x^2 y^5 z +
1008 a^8 b^2 c^8 x^2 y^5 z + 352 a^7 b^3 c^8 x^2 y^5 z +
352 a^6 b^4 c^8 x^2 y^5 z + 1520 a^5 b^5 c^8 x^2 y^5 z -
80 a^4 b^6 c^8 x^2 y^5 z - 240 a^8 b c^9 x^2 y^5 z -
1152 a^7 b^2 c^9 x^2 y^5 z + 416 a^6 b^3 c^9 x^2 y^5 z -
1024 a^5 b^4 c^9 x^2 y^5 z - 560 a^4 b^5 c^9 x^2 y^5 z -
240 a^7 b c^10 x^2 y^5 z + 528 a^6 b^2 c^10 x^2 y^5 z -
848 a^5 b^3 c^10 x^2 y^5 z + 48 a^4 b^4 c^10 x^2 y^5 z +
432 a^6 b c^11 x^2 y^5 z + 272 a^4 b^3 c^11 x^2 y^5 z -
240 a^5 b c^12 x^2 y^5 z - 48 a^4 b^2 c^12 x^2 y^5 z +
48 a^4 b c^13 x^2 y^5 z + 128 a^10 b^2 c^6 x y^6 z -
256 a^9 b^3 c^6 x y^6 z - 512 a^8 b^4 c^6 x y^6 z +
256 a^7 b^5 c^6 x y^6 z + 384 a^6 b^6 c^6 x y^6 z -
512 a^9 b^2 c^7 x y^6 z + 256 a^8 b^3 c^7 x y^6 z +
1024 a^7 b^4 c^7 x y^6 z + 256 a^6 b^5 c^7 x y^6 z +
768 a^8 b^2 c^8 x y^6 z + 256 a^7 b^3 c^8 x y^6 z -
512 a^6 b^4 c^8 x y^6 z - 512 a^7 b^2 c^9 x y^6 z -
256 a^6 b^3 c^9 x y^6 z + 128 a^6 b^2 c^10 x y^6 z +
64 a^6 b^8 c^4 x^6 z^2 - 128 a^4 b^10 c^4 x^6 z^2 +
64 a^2 b^12 c^4 x^6 z^2 + 256 a^5 b^8 c^5 x^6 z^2 +
256 a^4 b^9 c^5 x^6 z^2 - 256 a^3 b^10 c^5 x^6 z^2 -
256 a^2 b^11 c^5 x^6 z^2 - 128 a^4 b^8 c^6 x^6 z^2 +
512 a^3 b^9 c^6 x^6 z^2 + 384 a^2 b^10 c^6 x^6 z^2 -
256 a^3 b^8 c^7 x^6 z^2 - 256 a^2 b^9 c^7 x^6 z^2 +
64 a^2 b^8 c^8 x^6 z^2 + 80 a^8 b^6 c^4 x^5 y z^2 +
240 a^7 b^7 c^4 x^5 y z^2 - 80 a^6 b^8 c^4 x^5 y z^2 -
560 a^5 b^9 c^4 x^5 y z^2 + 48 a^4 b^10 c^4 x^5 y z^2 +
272 a^3 b^11 c^4 x^5 y z^2 - 48 a^2 b^12 c^4 x^5 y z^2 +
48 a b^13 c^4 x^5 y z^2 - 176 a^7 b^6 c^5 x^5 y z^2 +
768 a^6 b^7 c^5 x^5 y z^2 + 1520 a^5 b^8 c^5 x^5 y z^2 -
1024 a^4 b^9 c^5 x^5 y z^2 - 848 a^3 b^10 c^5 x^5 y z^2 -
240 a b^12 c^5 x^5 y z^2 + 1424 a^6 b^6 c^6 x^5 y z^2 +
2480 a^5 b^7 c^6 x^5 y z^2 + 352 a^4 b^8 c^6 x^5 y z^2 +
416 a^3 b^9 c^6 x^5 y z^2 + 528 a^2 b^10 c^6 x^5 y z^2 +
432 a b^11 c^6 x^5 y z^2 + 400 a^5 b^6 c^7 x^5 y z^2 +
2176 a^4 b^7 c^7 x^5 y z^2 + 352 a^3 b^8 c^7 x^5 y z^2 -
1152 a^2 b^9 c^7 x^5 y z^2 - 240 a b^10 c^7 x^5 y z^2 -
1552 a^4 b^6 c^8 x^5 y z^2 + 80 a^3 b^7 c^8 x^5 y z^2 +
1008 a^2 b^8 c^8 x^5 y z^2 - 240 a b^9 c^8 x^5 y z^2 -
272 a^3 b^6 c^9 x^5 y z^2 - 384 a^2 b^7 c^9 x^5 y z^2 +
432 a b^8 c^9 x^5 y z^2 + 48 a^2 b^6 c^10 x^5 y z^2 -
240 a b^7 c^10 x^5 y z^2 + 48 a b^6 c^11 x^5 y z^2 +
294 a^10 b^4 c^4 x^4 y^2 z^2 - 276 a^9 b^5 c^4 x^4 y^2 z^2 +
142 a^8 b^6 c^4 x^4 y^2 z^2 + 624 a^7 b^7 c^4 x^4 y^2 z^2 -
916 a^6 b^8 c^4 x^4 y^2 z^2 - 664 a^5 b^9 c^4 x^4 y^2 z^2 +
492 a^4 b^10 c^4 x^4 y^2 z^2 + 304 a^3 b^11 c^4 x^4 y^2 z^2 -
18 a^2 b^12 c^4 x^4 y^2 z^2 + 12 a b^13 c^4 x^4 y^2 z^2 +
6 b^14 c^4 x^4 y^2 z^2 - 276 a^9 b^4 c^5 x^4 y^2 z^2 -
388 a^8 b^5 c^5 x^4 y^2 z^2 + 816 a^7 b^6 c^5 x^4 y^2 z^2 +
1552 a^6 b^7 c^5 x^4 y^2 z^2 + 1032 a^5 b^8 c^5 x^4 y^2 z^2 -
1592 a^4 b^9 c^5 x^4 y^2 z^2 - 976 a^3 b^10 c^5 x^4 y^2 z^2 -
48 a^2 b^11 c^5 x^4 y^2 z^2 - 84 a b^12 c^5 x^4 y^2 z^2 -
36 b^13 c^5 x^4 y^2 z^2 + 142 a^8 b^4 c^6 x^4 y^2 z^2 +
816 a^7 b^5 c^6 x^4 y^2 z^2 + 8072 a^6 b^6 c^6 x^4 y^2 z^2 +
5776 a^5 b^7 c^6 x^4 y^2 z^2 - 876 a^4 b^8 c^6 x^4 y^2 z^2 +
1104 a^3 b^9 c^6 x^4 y^2 z^2 + 264 a^2 b^10 c^6 x^4 y^2 z^2 +
240 a b^11 c^6 x^4 y^2 z^2 + 78 b^12 c^6 x^4 y^2 z^2 +
624 a^7 b^4 c^7 x^4 y^2 z^2 + 1552 a^6 b^5 c^7 x^4 y^2 z^2 +
5776 a^5 b^6 c^7 x^4 y^2 z^2 + 3952 a^4 b^7 c^7 x^4 y^2 z^2 -
432 a^3 b^8 c^7 x^4 y^2 z^2 - 336 a^2 b^9 c^7 x^4 y^2 z^2 -
336 a b^10 c^7 x^4 y^2 z^2 - 48 b^11 c^7 x^4 y^2 z^2 -
916 a^6 b^4 c^8 x^4 y^2 z^2 + 1032 a^5 b^5 c^8 x^4 y^2 z^2 -
876 a^4 b^6 c^8 x^4 y^2 z^2 - 432 a^3 b^7 c^8 x^4 y^2 z^2 +
276 a^2 b^8 c^8 x^4 y^2 z^2 + 168 a b^9 c^8 x^4 y^2 z^2 -
84 b^10 c^8 x^4 y^2 z^2 - 664 a^5 b^4 c^9 x^4 y^2 z^2 -
1592 a^4 b^5 c^9 x^4 y^2 z^2 + 1104 a^3 b^6 c^9 x^4 y^2 z^2 -
336 a^2 b^7 c^9 x^4 y^2 z^2 + 168 a b^8 c^9 x^4 y^2 z^2 +
168 b^9 c^9 x^4 y^2 z^2 + 492 a^4 b^4 c^10 x^4 y^2 z^2 -
976 a^3 b^5 c^10 x^4 y^2 z^2 + 264 a^2 b^6 c^10 x^4 y^2 z^2 -
336 a b^7 c^10 x^4 y^2 z^2 - 84 b^8 c^10 x^4 y^2 z^2 +
304 a^3 b^4 c^11 x^4 y^2 z^2 - 48 a^2 b^5 c^11 x^4 y^2 z^2 +
240 a b^6 c^11 x^4 y^2 z^2 - 48 b^7 c^11 x^4 y^2 z^2 -
18 a^2 b^4 c^12 x^4 y^2 z^2 - 84 a b^5 c^12 x^4 y^2 z^2 +
78 b^6 c^12 x^4 y^2 z^2 + 12 a b^4 c^13 x^4 y^2 z^2 -
36 b^5 c^13 x^4 y^2 z^2 + 6 b^4 c^14 x^4 y^2 z^2 +
28 a^12 b^2 c^4 x^3 y^3 z^2 + 8 a^11 b^3 c^4 x^3 y^3 z^2 +
844 a^10 b^4 c^4 x^3 y^3 z^2 - 608 a^9 b^5 c^4 x^3 y^3 z^2 -
872 a^8 b^6 c^4 x^3 y^3 z^2 + 1200 a^7 b^7 c^4 x^3 y^3 z^2 -
872 a^6 b^8 c^4 x^3 y^3 z^2 - 608 a^5 b^9 c^4 x^3 y^3 z^2 +
844 a^4 b^10 c^4 x^3 y^3 z^2 + 8 a^3 b^11 c^4 x^3 y^3 z^2 +
28 a^2 b^12 c^4 x^3 y^3 z^2 - 136 a^11 b^2 c^5 x^3 y^3 z^2 -
216 a^10 b^3 c^5 x^3 y^3 z^2 - 1008 a^9 b^4 c^5 x^3 y^3 z^2 -
464 a^8 b^5 c^5 x^3 y^3 z^2 + 1824 a^7 b^6 c^5 x^3 y^3 z^2 +
1824 a^6 b^7 c^5 x^3 y^3 z^2 - 464 a^5 b^8 c^5 x^3 y^3 z^2 -
1008 a^4 b^9 c^5 x^3 y^3 z^2 - 216 a^3 b^10 c^5 x^3 y^3 z^2 -
136 a^2 b^11 c^5 x^3 y^3 z^2 + 204 a^10 b^2 c^6 x^3 y^3 z^2 +
624 a^9 b^3 c^6 x^3 y^3 z^2 - 848 a^8 b^4 c^6 x^3 y^3 z^2 +
4496 a^7 b^5 c^6 x^3 y^3 z^2 + 12552 a^6 b^6 c^6 x^3 y^3 z^2 +
4496 a^5 b^7 c^6 x^3 y^3 z^2 - 848 a^4 b^8 c^6 x^3 y^3 z^2 +
624 a^3 b^9 c^6 x^3 y^3 z^2 + 204 a^2 b^10 c^6 x^3 y^3 z^2 -
592 a^8 b^3 c^7 x^3 y^3 z^2 + 2736 a^7 b^4 c^7 x^3 y^3 z^2 +
6560 a^6 b^5 c^7 x^3 y^3 z^2 + 6560 a^5 b^6 c^7 x^3 y^3 z^2 +
2736 a^4 b^7 c^7 x^3 y^3 z^2 - 592 a^3 b^8 c^7 x^3 y^3 z^2 -
296 a^8 b^2 c^8 x^3 y^3 z^2 - 64 a^7 b^3 c^8 x^3 y^3 z^2 -
1208 a^6 b^4 c^8 x^3 y^3 z^2 + 3008 a^5 b^5 c^8 x^3 y^3 z^2 -
1208 a^4 b^6 c^8 x^3 y^3 z^2 - 64 a^3 b^7 c^8 x^3 y^3 z^2 -
296 a^2 b^8 c^8 x^3 y^3 z^2 + 304 a^7 b^2 c^9 x^3 y^3 z^2 +
704 a^6 b^3 c^9 x^3 y^3 z^2 - 1936 a^5 b^4 c^9 x^3 y^3 z^2 -
1936 a^4 b^5 c^9 x^3 y^3 z^2 + 704 a^3 b^6 c^9 x^3 y^3 z^2 +
304 a^2 b^7 c^9 x^3 y^3 z^2 - 104 a^6 b^2 c^10 x^3 y^3 z^2 -
752 a^5 b^3 c^10 x^3 y^3 z^2 + 1072 a^4 b^4 c^10 x^3 y^3 z^2 -
752 a^3 b^5 c^10 x^3 y^3 z^2 - 104 a^2 b^6 c^10 x^3 y^3 z^2 -
64 a^5 b^2 c^11 x^3 y^3 z^2 + 208 a^4 b^3 c^11 x^3 y^3 z^2 +
208 a^3 b^4 c^11 x^3 y^3 z^2 - 64 a^2 b^5 c^11 x^3 y^3 z^2 +
140 a^4 b^2 c^12 x^3 y^3 z^2 + 184 a^3 b^3 c^12 x^3 y^3 z^2 +
140 a^2 b^4 c^12 x^3 y^3 z^2 - 104 a^3 b^2 c^13 x^3 y^3 z^2 -
104 a^2 b^3 c^13 x^3 y^3 z^2 + 28 a^2 b^2 c^14 x^3 y^3 z^2 +
6 a^14 c^4 x^2 y^4 z^2 + 12 a^13 b c^4 x^2 y^4 z^2 -
18 a^12 b^2 c^4 x^2 y^4 z^2 + 304 a^11 b^3 c^4 x^2 y^4 z^2 +
492 a^10 b^4 c^4 x^2 y^4 z^2 - 664 a^9 b^5 c^4 x^2 y^4 z^2 -
916 a^8 b^6 c^4 x^2 y^4 z^2 + 624 a^7 b^7 c^4 x^2 y^4 z^2 +
142 a^6 b^8 c^4 x^2 y^4 z^2 - 276 a^5 b^9 c^4 x^2 y^4 z^2 +
294 a^4 b^10 c^4 x^2 y^4 z^2 - 36 a^13 c^5 x^2 y^4 z^2 -
84 a^12 b c^5 x^2 y^4 z^2 - 48 a^11 b^2 c^5 x^2 y^4 z^2 -
976 a^10 b^3 c^5 x^2 y^4 z^2 - 1592 a^9 b^4 c^5 x^2 y^4 z^2 +
1032 a^8 b^5 c^5 x^2 y^4 z^2 + 1552 a^7 b^6 c^5 x^2 y^4 z^2 +
816 a^6 b^7 c^5 x^2 y^4 z^2 - 388 a^5 b^8 c^5 x^2 y^4 z^2 -
276 a^4 b^9 c^5 x^2 y^4 z^2 + 78 a^12 c^6 x^2 y^4 z^2 +
240 a^11 b c^6 x^2 y^4 z^2 + 264 a^10 b^2 c^6 x^2 y^4 z^2 +
1104 a^9 b^3 c^6 x^2 y^4 z^2 - 876 a^8 b^4 c^6 x^2 y^4 z^2 +
5776 a^7 b^5 c^6 x^2 y^4 z^2 + 8072 a^6 b^6 c^6 x^2 y^4 z^2 +
816 a^5 b^7 c^6 x^2 y^4 z^2 + 142 a^4 b^8 c^6 x^2 y^4 z^2 -
48 a^11 c^7 x^2 y^4 z^2 - 336 a^10 b c^7 x^2 y^4 z^2 -
336 a^9 b^2 c^7 x^2 y^4 z^2 - 432 a^8 b^3 c^7 x^2 y^4 z^2 +
3952 a^7 b^4 c^7 x^2 y^4 z^2 + 5776 a^6 b^5 c^7 x^2 y^4 z^2 +
1552 a^5 b^6 c^7 x^2 y^4 z^2 + 624 a^4 b^7 c^7 x^2 y^4 z^2 -
84 a^10 c^8 x^2 y^4 z^2 + 168 a^9 b c^8 x^2 y^4 z^2 +
276 a^8 b^2 c^8 x^2 y^4 z^2 - 432 a^7 b^3 c^8 x^2 y^4 z^2 -
876 a^6 b^4 c^8 x^2 y^4 z^2 + 1032 a^5 b^5 c^8 x^2 y^4 z^2 -
916 a^4 b^6 c^8 x^2 y^4 z^2 + 168 a^9 c^9 x^2 y^4 z^2 +
168 a^8 b c^9 x^2 y^4 z^2 - 336 a^7 b^2 c^9 x^2 y^4 z^2 +
1104 a^6 b^3 c^9 x^2 y^4 z^2 - 1592 a^5 b^4 c^9 x^2 y^4 z^2 -
664 a^4 b^5 c^9 x^2 y^4 z^2 - 84 a^8 c^10 x^2 y^4 z^2 -
336 a^7 b c^10 x^2 y^4 z^2 + 264 a^6 b^2 c^10 x^2 y^4 z^2 -
976 a^5 b^3 c^10 x^2 y^4 z^2 + 492 a^4 b^4 c^10 x^2 y^4 z^2 -
48 a^7 c^11 x^2 y^4 z^2 + 240 a^6 b c^11 x^2 y^4 z^2 -
48 a^5 b^2 c^11 x^2 y^4 z^2 + 304 a^4 b^3 c^11 x^2 y^4 z^2 +
78 a^6 c^12 x^2 y^4 z^2 - 84 a^5 b c^12 x^2 y^4 z^2 -
18 a^4 b^2 c^12 x^2 y^4 z^2 - 36 a^5 c^13 x^2 y^4 z^2 +
12 a^4 b c^13 x^2 y^4 z^2 + 6 a^4 c^14 x^2 y^4 z^2 +
48 a^13 b c^4 x y^5 z^2 - 48 a^12 b^2 c^4 x y^5 z^2 +
272 a^11 b^3 c^4 x y^5 z^2 + 48 a^10 b^4 c^4 x y^5 z^2 -
560 a^9 b^5 c^4 x y^5 z^2 - 80 a^8 b^6 c^4 x y^5 z^2 +
240 a^7 b^7 c^4 x y^5 z^2 + 80 a^6 b^8 c^4 x y^5 z^2 -
240 a^12 b c^5 x y^5 z^2 - 848 a^10 b^3 c^5 x y^5 z^2 -
1024 a^9 b^4 c^5 x y^5 z^2 + 1520 a^8 b^5 c^5 x y^5 z^2 +
768 a^7 b^6 c^5 x y^5 z^2 - 176 a^6 b^7 c^5 x y^5 z^2 +
432 a^11 b c^6 x y^5 z^2 + 528 a^10 b^2 c^6 x y^5 z^2 +
416 a^9 b^3 c^6 x y^5 z^2 + 352 a^8 b^4 c^6 x y^5 z^2 +
2480 a^7 b^5 c^6 x y^5 z^2 + 1424 a^6 b^6 c^6 x y^5 z^2 -
240 a^10 b c^7 x y^5 z^2 - 1152 a^9 b^2 c^7 x y^5 z^2 +
352 a^8 b^3 c^7 x y^5 z^2 + 2176 a^7 b^4 c^7 x y^5 z^2 +
400 a^6 b^5 c^7 x y^5 z^2 - 240 a^9 b c^8 x y^5 z^2 +
1008 a^8 b^2 c^8 x y^5 z^2 + 80 a^7 b^3 c^8 x y^5 z^2 -
1552 a^6 b^4 c^8 x y^5 z^2 + 432 a^8 b c^9 x y^5 z^2 -
384 a^7 b^2 c^9 x y^5 z^2 - 272 a^6 b^3 c^9 x y^5 z^2 -
240 a^7 b c^10 x y^5 z^2 + 48 a^6 b^2 c^10 x y^5 z^2 +
48 a^6 b c^11 x y^5 z^2 + 64 a^12 b^2 c^4 y^6 z^2 -
128 a^10 b^4 c^4 y^6 z^2 + 64 a^8 b^6 c^4 y^6 z^2 -
256 a^11 b^2 c^5 y^6 z^2 - 256 a^10 b^3 c^5 y^6 z^2 +
256 a^9 b^4 c^5 y^6 z^2 + 256 a^8 b^5 c^5 y^6 z^2 +
384 a^10 b^2 c^6 y^6 z^2 + 512 a^9 b^3 c^6 y^6 z^2 -
128 a^8 b^4 c^6 y^6 z^2 - 256 a^9 b^2 c^7 y^6 z^2 -
256 a^8 b^3 c^7 y^6 z^2 + 64 a^8 b^2 c^8 y^6 z^2 -
16 a^8 b^8 c^2 x^5 z^3 + 48 a^7 b^9 c^2 x^5 z^3 -
16 a^6 b^10 c^2 x^5 z^3 - 80 a^5 b^11 c^2 x^5 z^3 +
80 a^4 b^12 c^2 x^5 z^3 + 16 a^3 b^13 c^2 x^5 z^3 -
48 a^2 b^14 c^2 x^5 z^3 + 16 a b^15 c^2 x^5 z^3 -
112 a^7 b^8 c^3 x^5 z^3 + 256 a^6 b^9 c^3 x^5 z^3 +
16 a^5 b^10 c^3 x^5 z^3 - 384 a^4 b^11 c^3 x^5 z^3 +
176 a^3 b^12 c^3 x^5 z^3 + 128 a^2 b^13 c^3 x^5 z^3 -
80 a b^14 c^3 x^5 z^3 - 48 a^6 b^8 c^4 x^5 z^3 +
208 a^5 b^9 c^4 x^5 z^3 + 32 a^4 b^10 c^4 x^5 z^3 -
352 a^3 b^11 c^4 x^5 z^3 + 16 a^2 b^12 c^4 x^5 z^3 +
144 a b^13 c^4 x^5 z^3 + 1136 a^5 b^8 c^5 x^5 z^3 +
768 a^4 b^9 c^5 x^5 z^3 - 416 a^3 b^10 c^5 x^5 z^3 -
384 a^2 b^11 c^5 x^5 z^3 - 80 a b^12 c^5 x^5 z^3 -
496 a^4 b^8 c^6 x^5 z^3 + 1104 a^3 b^9 c^6 x^5 z^3 +
496 a^2 b^10 c^6 x^5 z^3 - 80 a b^11 c^6 x^5 z^3 -
528 a^3 b^8 c^7 x^5 z^3 - 256 a^2 b^9 c^7 x^5 z^3 +
144 a b^10 c^7 x^5 z^3 + 48 a^2 b^8 c^8 x^5 z^3 -
80 a b^9 c^8 x^5 z^3 + 16 a b^8 c^9 x^5 z^3 -
12 a^10 b^6 c^2 x^4 y z^3 + 8 a^9 b^7 c^2 x^4 y z^3 +
68 a^8 b^8 c^2 x^4 y z^3 - 80 a^7 b^9 c^2 x^4 y z^3 -
72 a^6 b^10 c^2 x^4 y z^3 + 128 a^5 b^11 c^2 x^4 y z^3 -
8 a^4 b^12 c^2 x^4 y z^3 - 48 a^3 b^13 c^2 x^4 y z^3 +
20 a^2 b^14 c^2 x^4 y z^3 - 8 a b^15 c^2 x^4 y z^3 +
4 b^16 c^2 x^4 y z^3 - 24 a^9 b^6 c^3 x^4 y z^3 -
120 a^8 b^7 c^3 x^4 y z^3 + 368 a^7 b^8 c^3 x^4 y z^3 -
512 a^5 b^10 c^3 x^4 y z^3 + 208 a^4 b^11 c^3 x^4 y z^3 +
144 a^3 b^12 c^3 x^4 y z^3 - 64 a^2 b^13 c^3 x^4 y z^3 +
24 a b^14 c^3 x^4 y z^3 - 24 b^15 c^3 x^4 y z^3 -
316 a^8 b^6 c^4 x^4 y z^3 + 1072 a^7 b^7 c^4 x^4 y z^3 +
144 a^6 b^8 c^4 x^4 y z^3 - 1536 a^5 b^9 c^4 x^4 y z^3 +
136 a^4 b^10 c^4 x^4 y z^3 + 432 a^3 b^11 c^4 x^4 y z^3 -
16 a^2 b^12 c^4 x^4 y z^3 + 32 a b^13 c^4 x^4 y z^3 +
52 b^14 c^4 x^4 y z^3 - 208 a^7 b^6 c^5 x^4 y z^3 +
3136 a^6 b^7 c^5 x^4 y z^3 + 5376 a^5 b^8 c^5 x^4 y z^3 -
1824 a^4 b^9 c^5 x^4 y z^3 - 1232 a^3 b^10 c^5 x^4 y z^3 +
128 a^2 b^11 c^5 x^4 y z^3 - 224 a b^12 c^5 x^4 y z^3 -
32 b^13 c^5 x^4 y z^3 + 1912 a^6 b^6 c^6 x^4 y z^3 +
4480 a^5 b^7 c^6 x^4 y z^3 + 904 a^4 b^8 c^6 x^4 y z^3 +
432 a^3 b^9 c^6 x^4 y z^3 + 184 a^2 b^10 c^6 x^4 y z^3 +
336 a b^11 c^6 x^4 y z^3 - 56 b^12 c^6 x^4 y z^3 +
256 a^5 b^6 c^7 x^4 y z^3 + 2128 a^4 b^7 c^7 x^4 y z^3 +
624 a^3 b^8 c^7 x^4 y z^3 - 576 a^2 b^9 c^7 x^4 y z^3 -
112 a b^10 c^7 x^4 y z^3 + 112 b^11 c^7 x^4 y z^3 -
1544 a^4 b^6 c^8 x^4 y z^3 - 304 a^3 b^7 c^8 x^4 y z^3 +
368 a^2 b^8 c^8 x^4 y z^3 - 224 a b^9 c^8 x^4 y z^3 -
56 b^10 c^8 x^4 y z^3 - 48 a^3 b^6 c^9 x^4 y z^3 +
288 a b^8 c^9 x^4 y z^3 - 32 b^9 c^9 x^4 y z^3 -
44 a^2 b^6 c^10 x^4 y z^3 - 136 a b^7 c^10 x^4 y z^3 +
52 b^8 c^10 x^4 y z^3 + 24 a b^6 c^11 x^4 y z^3 -
24 b^7 c^11 x^4 y z^3 + 4 b^6 c^12 x^4 y z^3 +
28 a^12 b^4 c^2 x^3 y^2 z^3 - 136 a^11 b^5 c^2 x^3 y^2 z^3 +
204 a^10 b^6 c^2 x^3 y^2 z^3 - 296 a^8 b^8 c^2 x^3 y^2 z^3 +
304 a^7 b^9 c^2 x^3 y^2 z^3 - 104 a^6 b^10 c^2 x^3 y^2 z^3 -
64 a^5 b^11 c^2 x^3 y^2 z^3 + 140 a^4 b^12 c^2 x^3 y^2 z^3 -
104 a^3 b^13 c^2 x^3 y^2 z^3 + 28 a^2 b^14 c^2 x^3 y^2 z^3 +
8 a^11 b^4 c^3 x^3 y^2 z^3 - 216 a^10 b^5 c^3 x^3 y^2 z^3 +
624 a^9 b^6 c^3 x^3 y^2 z^3 - 592 a^8 b^7 c^3 x^3 y^2 z^3 -
64 a^7 b^8 c^3 x^3 y^2 z^3 + 704 a^6 b^9 c^3 x^3 y^2 z^3 -
752 a^5 b^10 c^3 x^3 y^2 z^3 + 208 a^4 b^11 c^3 x^3 y^2 z^3 +
184 a^3 b^12 c^3 x^3 y^2 z^3 - 104 a^2 b^13 c^3 x^3 y^2 z^3 +
844 a^10 b^4 c^4 x^3 y^2 z^3 - 1008 a^9 b^5 c^4 x^3 y^2 z^3 -
848 a^8 b^6 c^4 x^3 y^2 z^3 + 2736 a^7 b^7 c^4 x^3 y^2 z^3 -
1208 a^6 b^8 c^4 x^3 y^2 z^3 - 1936 a^5 b^9 c^4 x^3 y^2 z^3 +
1072 a^4 b^10 c^4 x^3 y^2 z^3 + 208 a^3 b^11 c^4 x^3 y^2 z^3 +
140 a^2 b^12 c^4 x^3 y^2 z^3 - 608 a^9 b^4 c^5 x^3 y^2 z^3 -
464 a^8 b^5 c^5 x^3 y^2 z^3 + 4496 a^7 b^6 c^5 x^3 y^2 z^3 +
6560 a^6 b^7 c^5 x^3 y^2 z^3 + 3008 a^5 b^8 c^5 x^3 y^2 z^3 -
1936 a^4 b^9 c^5 x^3 y^2 z^3 - 752 a^3 b^10 c^5 x^3 y^2 z^3 -
64 a^2 b^11 c^5 x^3 y^2 z^3 - 872 a^8 b^4 c^6 x^3 y^2 z^3 +
1824 a^7 b^5 c^6 x^3 y^2 z^3 + 12552 a^6 b^6 c^6 x^3 y^2 z^3 +
6560 a^5 b^7 c^6 x^3 y^2 z^3 - 1208 a^4 b^8 c^6 x^3 y^2 z^3 +
704 a^3 b^9 c^6 x^3 y^2 z^3 - 104 a^2 b^10 c^6 x^3 y^2 z^3 +
1200 a^7 b^4 c^7 x^3 y^2 z^3 + 1824 a^6 b^5 c^7 x^3 y^2 z^3 +
4496 a^5 b^6 c^7 x^3 y^2 z^3 + 2736 a^4 b^7 c^7 x^3 y^2 z^3 -
64 a^3 b^8 c^7 x^3 y^2 z^3 + 304 a^2 b^9 c^7 x^3 y^2 z^3 -
872 a^6 b^4 c^8 x^3 y^2 z^3 - 464 a^5 b^5 c^8 x^3 y^2 z^3 -
848 a^4 b^6 c^8 x^3 y^2 z^3 - 592 a^3 b^7 c^8 x^3 y^2 z^3 -
296 a^2 b^8 c^8 x^3 y^2 z^3 - 608 a^5 b^4 c^9 x^3 y^2 z^3 -
1008 a^4 b^5 c^9 x^3 y^2 z^3 + 624 a^3 b^6 c^9 x^3 y^2 z^3 +
844 a^4 b^4 c^10 x^3 y^2 z^3 - 216 a^3 b^5 c^10 x^3 y^2 z^3 +
204 a^2 b^6 c^10 x^3 y^2 z^3 + 8 a^3 b^4 c^11 x^3 y^2 z^3 -
136 a^2 b^5 c^11 x^3 y^2 z^3 + 28 a^2 b^4 c^12 x^3 y^2 z^3 +
28 a^14 b^2 c^2 x^2 y^3 z^3 - 104 a^13 b^3 c^2 x^2 y^3 z^3 +
140 a^12 b^4 c^2 x^2 y^3 z^3 - 64 a^11 b^5 c^2 x^2 y^3 z^3 -
104 a^10 b^6 c^2 x^2 y^3 z^3 + 304 a^9 b^7 c^2 x^2 y^3 z^3 -
296 a^8 b^8 c^2 x^2 y^3 z^3 + 204 a^6 b^10 c^2 x^2 y^3 z^3 -
136 a^5 b^11 c^2 x^2 y^3 z^3 + 28 a^4 b^12 c^2 x^2 y^3 z^3 -
104 a^13 b^2 c^3 x^2 y^3 z^3 + 184 a^12 b^3 c^3 x^2 y^3 z^3 +
208 a^11 b^4 c^3 x^2 y^3 z^3 - 752 a^10 b^5 c^3 x^2 y^3 z^3 +
704 a^9 b^6 c^3 x^2 y^3 z^3 - 64 a^8 b^7 c^3 x^2 y^3 z^3 -
592 a^7 b^8 c^3 x^2 y^3 z^3 + 624 a^6 b^9 c^3 x^2 y^3 z^3 -
216 a^5 b^10 c^3 x^2 y^3 z^3 + 8 a^4 b^11 c^3 x^2 y^3 z^3 +
140 a^12 b^2 c^4 x^2 y^3 z^3 + 208 a^11 b^3 c^4 x^2 y^3 z^3 +
1072 a^10 b^4 c^4 x^2 y^3 z^3 - 1936 a^9 b^5 c^4 x^2 y^3 z^3 -
1208 a^8 b^6 c^4 x^2 y^3 z^3 + 2736 a^7 b^7 c^4 x^2 y^3 z^3 -
848 a^6 b^8 c^4 x^2 y^3 z^3 - 1008 a^5 b^9 c^4 x^2 y^3 z^3 +
844 a^4 b^10 c^4 x^2 y^3 z^3 - 64 a^11 b^2 c^5 x^2 y^3 z^3 -
752 a^10 b^3 c^5 x^2 y^3 z^3 - 1936 a^9 b^4 c^5 x^2 y^3 z^3 +
3008 a^8 b^5 c^5 x^2 y^3 z^3 + 6560 a^7 b^6 c^5 x^2 y^3 z^3 +
4496 a^6 b^7 c^5 x^2 y^3 z^3 - 464 a^5 b^8 c^5 x^2 y^3 z^3 -
608 a^4 b^9 c^5 x^2 y^3 z^3 - 104 a^10 b^2 c^6 x^2 y^3 z^3 +
704 a^9 b^3 c^6 x^2 y^3 z^3 - 1208 a^8 b^4 c^6 x^2 y^3 z^3 +
6560 a^7 b^5 c^6 x^2 y^3 z^3 + 12552 a^6 b^6 c^6 x^2 y^3 z^3 +
1824 a^5 b^7 c^6 x^2 y^3 z^3 - 872 a^4 b^8 c^6 x^2 y^3 z^3 +
304 a^9 b^2 c^7 x^2 y^3 z^3 - 64 a^8 b^3 c^7 x^2 y^3 z^3 +
2736 a^7 b^4 c^7 x^2 y^3 z^3 + 4496 a^6 b^5 c^7 x^2 y^3 z^3 +
1824 a^5 b^6 c^7 x^2 y^3 z^3 + 1200 a^4 b^7 c^7 x^2 y^3 z^3 -
296 a^8 b^2 c^8 x^2 y^3 z^3 - 592 a^7 b^3 c^8 x^2 y^3 z^3 -
848 a^6 b^4 c^8 x^2 y^3 z^3 - 464 a^5 b^5 c^8 x^2 y^3 z^3 -
872 a^4 b^6 c^8 x^2 y^3 z^3 + 624 a^6 b^3 c^9 x^2 y^3 z^3 -
1008 a^5 b^4 c^9 x^2 y^3 z^3 - 608 a^4 b^5 c^9 x^2 y^3 z^3 +
204 a^6 b^2 c^10 x^2 y^3 z^3 - 216 a^5 b^3 c^10 x^2 y^3 z^3 +
844 a^4 b^4 c^10 x^2 y^3 z^3 - 136 a^5 b^2 c^11 x^2 y^3 z^3 +
8 a^4 b^3 c^11 x^2 y^3 z^3 + 28 a^4 b^2 c^12 x^2 y^3 z^3 +
4 a^16 c^2 x y^4 z^3 - 8 a^15 b c^2 x y^4 z^3 +
20 a^14 b^2 c^2 x y^4 z^3 - 48 a^13 b^3 c^2 x y^4 z^3 -
8 a^12 b^4 c^2 x y^4 z^3 + 128 a^11 b^5 c^2 x y^4 z^3 -
72 a^10 b^6 c^2 x y^4 z^3 - 80 a^9 b^7 c^2 x y^4 z^3 +
68 a^8 b^8 c^2 x y^4 z^3 + 8 a^7 b^9 c^2 x y^4 z^3 -
12 a^6 b^10 c^2 x y^4 z^3 - 24 a^15 c^3 x y^4 z^3 +
24 a^14 b c^3 x y^4 z^3 - 64 a^13 b^2 c^3 x y^4 z^3 +
144 a^12 b^3 c^3 x y^4 z^3 + 208 a^11 b^4 c^3 x y^4 z^3 -
512 a^10 b^5 c^3 x y^4 z^3 + 368 a^8 b^7 c^3 x y^4 z^3 -
120 a^7 b^8 c^3 x y^4 z^3 - 24 a^6 b^9 c^3 x y^4 z^3 +
52 a^14 c^4 x y^4 z^3 + 32 a^13 b c^4 x y^4 z^3 -
16 a^12 b^2 c^4 x y^4 z^3 + 432 a^11 b^3 c^4 x y^4 z^3 +
136 a^10 b^4 c^4 x y^4 z^3 - 1536 a^9 b^5 c^4 x y^4 z^3 +
144 a^8 b^6 c^4 x y^4 z^3 + 1072 a^7 b^7 c^4 x y^4 z^3 -
316 a^6 b^8 c^4 x y^4 z^3 - 32 a^13 c^5 x y^4 z^3 -
224 a^12 b c^5 x y^4 z^3 + 128 a^11 b^2 c^5 x y^4 z^3 -
1232 a^10 b^3 c^5 x y^4 z^3 - 1824 a^9 b^4 c^5 x y^4 z^3 +
5376 a^8 b^5 c^5 x y^4 z^3 + 3136 a^7 b^6 c^5 x y^4 z^3 -
208 a^6 b^7 c^5 x y^4 z^3 - 56 a^12 c^6 x y^4 z^3 +
336 a^11 b c^6 x y^4 z^3 + 184 a^10 b^2 c^6 x y^4 z^3 +
432 a^9 b^3 c^6 x y^4 z^3 + 904 a^8 b^4 c^6 x y^4 z^3 +
4480 a^7 b^5 c^6 x y^4 z^3 + 1912 a^6 b^6 c^6 x y^4 z^3 +
112 a^11 c^7 x y^4 z^3 - 112 a^10 b c^7 x y^4 z^3 -
576 a^9 b^2 c^7 x y^4 z^3 + 624 a^8 b^3 c^7 x y^4 z^3 +
2128 a^7 b^4 c^7 x y^4 z^3 + 256 a^6 b^5 c^7 x y^4 z^3 -
56 a^10 c^8 x y^4 z^3 - 224 a^9 b c^8 x y^4 z^3 +
368 a^8 b^2 c^8 x y^4 z^3 - 304 a^7 b^3 c^8 x y^4 z^3 -
1544 a^6 b^4 c^8 x y^4 z^3 - 32 a^9 c^9 x y^4 z^3 +
288 a^8 b c^9 x y^4 z^3 - 48 a^6 b^3 c^9 x y^4 z^3 +
52 a^8 c^10 x y^4 z^3 - 136 a^7 b c^10 x y^4 z^3 -
44 a^6 b^2 c^10 x y^4 z^3 - 24 a^7 c^11 x y^4 z^3 +
24 a^6 b c^11 x y^4 z^3 + 4 a^6 c^12 x y^4 z^3 +
16 a^15 b c^2 y^5 z^3 - 48 a^14 b^2 c^2 y^5 z^3 +
16 a^13 b^3 c^2 y^5 z^3 + 80 a^12 b^4 c^2 y^5 z^3 -
80 a^11 b^5 c^2 y^5 z^3 - 16 a^10 b^6 c^2 y^5 z^3 +
48 a^9 b^7 c^2 y^5 z^3 - 16 a^8 b^8 c^2 y^5 z^3 -
80 a^14 b c^3 y^5 z^3 + 128 a^13 b^2 c^3 y^5 z^3 +
176 a^12 b^3 c^3 y^5 z^3 - 384 a^11 b^4 c^3 y^5 z^3 +
16 a^10 b^5 c^3 y^5 z^3 + 256 a^9 b^6 c^3 y^5 z^3 -
112 a^8 b^7 c^3 y^5 z^3 + 144 a^13 b c^4 y^5 z^3 +
16 a^12 b^2 c^4 y^5 z^3 - 352 a^11 b^3 c^4 y^5 z^3 +
32 a^10 b^4 c^4 y^5 z^3 + 208 a^9 b^5 c^4 y^5 z^3 -
48 a^8 b^6 c^4 y^5 z^3 - 80 a^12 b c^5 y^5 z^3 -
384 a^11 b^2 c^5 y^5 z^3 - 416 a^10 b^3 c^5 y^5 z^3 +
768 a^9 b^4 c^5 y^5 z^3 + 1136 a^8 b^5 c^5 y^5 z^3 -
80 a^11 b c^6 y^5 z^3 + 496 a^10 b^2 c^6 y^5 z^3 +
1104 a^9 b^3 c^6 y^5 z^3 - 496 a^8 b^4 c^6 y^5 z^3 +
144 a^10 b c^7 y^5 z^3 - 256 a^9 b^2 c^7 y^5 z^3 -
528 a^8 b^3 c^7 y^5 z^3 - 80 a^9 b c^8 y^5 z^3 +
48 a^8 b^2 c^8 y^5 z^3 + 16 a^8 b c^9 y^5 z^3 + a^10 b^8 x^4 z^4 -
6 a^9 b^9 x^4 z^4 + 13 a^8 b^10 x^4 z^4 - 8 a^7 b^11 x^4 z^4 -
14 a^6 b^12 x^4 z^4 + 28 a^5 b^13 x^4 z^4 - 14 a^4 b^14 x^4 z^4 -
8 a^3 b^15 x^4 z^4 + 13 a^2 b^16 x^4 z^4 - 6 a b^17 x^4 z^4 +
b^18 x^4 z^4 + 10 a^9 b^8 c x^4 z^4 - 54 a^8 b^9 c x^4 z^4 +
104 a^7 b^10 c x^4 z^4 - 56 a^6 b^11 c x^4 z^4 -
84 a^5 b^12 c x^4 z^4 + 140 a^4 b^13 c x^4 z^4 -
56 a^3 b^14 c x^4 z^4 - 24 a^2 b^15 c x^4 z^4 +
26 a b^16 c x^4 z^4 - 6 b^17 c x^4 z^4 - 19 a^8 b^8 c^2 x^4 z^4 +
8 a^7 b^9 c^2 x^4 z^4 + 140 a^6 b^10 c^2 x^4 z^4 -
232 a^5 b^11 c^2 x^4 z^4 + 46 a^4 b^12 c^2 x^4 z^4 +
120 a^3 b^13 c^2 x^4 z^4 - 52 a^2 b^14 c^2 x^4 z^4 -
24 a b^15 c^2 x^4 z^4 + 13 b^16 c^2 x^4 z^4 -
392 a^7 b^8 c^3 x^4 z^4 + 872 a^6 b^9 c^3 x^4 z^4 -
168 a^5 b^10 c^3 x^4 z^4 - 728 a^4 b^11 c^3 x^4 z^4 +
360 a^3 b^12 c^3 x^4 z^4 + 120 a^2 b^13 c^3 x^4 z^4 -
56 a b^14 c^3 x^4 z^4 - 8 b^15 c^3 x^4 z^4 -
494 a^6 b^8 c^4 x^4 z^4 + 716 a^5 b^9 c^4 x^4 z^4 +
334 a^4 b^10 c^4 x^4 z^4 - 728 a^3 b^11 c^4 x^4 z^4 +
46 a^2 b^12 c^4 x^4 z^4 + 140 a b^13 c^4 x^4 z^4 -
14 b^14 c^4 x^4 z^4 + 1788 a^5 b^8 c^5 x^4 z^4 +
716 a^4 b^9 c^5 x^4 z^4 - 168 a^3 b^10 c^5 x^4 z^4 -
232 a^2 b^11 c^5 x^4 z^4 - 84 a b^12 c^5 x^4 z^4 +
28 b^13 c^5 x^4 z^4 - 494 a^4 b^8 c^6 x^4 z^4 +
872 a^3 b^9 c^6 x^4 z^4 + 140 a^2 b^10 c^6 x^4 z^4 -
56 a b^11 c^6 x^4 z^4 - 14 b^12 c^6 x^4 z^4 -
392 a^3 b^8 c^7 x^4 z^4 + 8 a^2 b^9 c^7 x^4 z^4 +
104 a b^10 c^7 x^4 z^4 - 8 b^11 c^7 x^4 z^4 -
19 a^2 b^8 c^8 x^4 z^4 - 54 a b^9 c^8 x^4 z^4 +
13 b^10 c^8 x^4 z^4 + 10 a b^8 c^9 x^4 z^4 - 6 b^9 c^9 x^4 z^4 +
b^8 c^10 x^4 z^4 + 4 a^12 b^6 x^3 y z^4 - 24 a^11 b^7 x^3 y z^4 +
52 a^10 b^8 x^3 y z^4 - 32 a^9 b^9 x^3 y z^4 -
56 a^8 b^10 x^3 y z^4 + 112 a^7 b^11 x^3 y z^4 -
56 a^6 b^12 x^3 y z^4 - 32 a^5 b^13 x^3 y z^4 +
52 a^4 b^14 x^3 y z^4 - 24 a^3 b^15 x^3 y z^4 +
4 a^2 b^16 x^3 y z^4 + 24 a^11 b^6 c x^3 y z^4 -
136 a^10 b^7 c x^3 y z^4 + 288 a^9 b^8 c x^3 y z^4 -
224 a^8 b^9 c x^3 y z^4 - 112 a^7 b^10 c x^3 y z^4 +
336 a^6 b^11 c x^3 y z^4 - 224 a^5 b^12 c x^3 y z^4 +
32 a^4 b^13 c x^3 y z^4 + 24 a^3 b^14 c x^3 y z^4 -
8 a^2 b^15 c x^3 y z^4 - 44 a^10 b^6 c^2 x^3 y z^4 +
368 a^8 b^8 c^2 x^3 y z^4 - 576 a^7 b^9 c^2 x^3 y z^4 +
184 a^6 b^10 c^2 x^3 y z^4 + 128 a^5 b^11 c^2 x^3 y z^4 -
16 a^4 b^12 c^2 x^3 y z^4 - 64 a^3 b^13 c^2 x^3 y z^4 +
20 a^2 b^14 c^2 x^3 y z^4 - 48 a^9 b^6 c^3 x^3 y z^4 -
304 a^8 b^7 c^3 x^3 y z^4 + 624 a^7 b^8 c^3 x^3 y z^4 +
432 a^6 b^9 c^3 x^3 y z^4 - 1232 a^5 b^10 c^3 x^3 y z^4 +
432 a^4 b^11 c^3 x^3 y z^4 + 144 a^3 b^12 c^3 x^3 y z^4 -
48 a^2 b^13 c^3 x^3 y z^4 - 1544 a^8 b^6 c^4 x^3 y z^4 +
2128 a^7 b^7 c^4 x^3 y z^4 + 904 a^6 b^8 c^4 x^3 y z^4 -
1824 a^5 b^9 c^4 x^3 y z^4 + 136 a^4 b^10 c^4 x^3 y z^4 +
208 a^3 b^11 c^4 x^3 y z^4 - 8 a^2 b^12 c^4 x^3 y z^4 +
256 a^7 b^6 c^5 x^3 y z^4 + 4480 a^6 b^7 c^5 x^3 y z^4 +
5376 a^5 b^8 c^5 x^3 y z^4 - 1536 a^4 b^9 c^5 x^3 y z^4 -
512 a^3 b^10 c^5 x^3 y z^4 + 128 a^2 b^11 c^5 x^3 y z^4 +
1912 a^6 b^6 c^6 x^3 y z^4 + 3136 a^5 b^7 c^6 x^3 y z^4 +
144 a^4 b^8 c^6 x^3 y z^4 - 72 a^2 b^10 c^6 x^3 y z^4 -
208 a^5 b^6 c^7 x^3 y z^4 + 1072 a^4 b^7 c^7 x^3 y z^4 +
368 a^3 b^8 c^7 x^3 y z^4 - 80 a^2 b^9 c^7 x^3 y z^4 -
316 a^4 b^6 c^8 x^3 y z^4 - 120 a^3 b^7 c^8 x^3 y z^4 +
68 a^2 b^8 c^8 x^3 y z^4 - 24 a^3 b^6 c^9 x^3 y z^4 +
8 a^2 b^7 c^9 x^3 y z^4 - 12 a^2 b^6 c^10 x^3 y z^4 +
6 a^14 b^4 x^2 y^2 z^4 - 36 a^13 b^5 x^2 y^2 z^4 +
78 a^12 b^6 x^2 y^2 z^4 - 48 a^11 b^7 x^2 y^2 z^4 -
84 a^10 b^8 x^2 y^2 z^4 + 168 a^9 b^9 x^2 y^2 z^4 -
84 a^8 b^10 x^2 y^2 z^4 - 48 a^7 b^11 x^2 y^2 z^4 +
78 a^6 b^12 x^2 y^2 z^4 - 36 a^5 b^13 x^2 y^2 z^4 +
6 a^4 b^14 x^2 y^2 z^4 + 12 a^13 b^4 c x^2 y^2 z^4 -
84 a^12 b^5 c x^2 y^2 z^4 + 240 a^11 b^6 c x^2 y^2 z^4 -
336 a^10 b^7 c x^2 y^2 z^4 + 168 a^9 b^8 c x^2 y^2 z^4 +
168 a^8 b^9 c x^2 y^2 z^4 - 336 a^7 b^10 c x^2 y^2 z^4 +
240 a^6 b^11 c x^2 y^2 z^4 - 84 a^5 b^12 c x^2 y^2 z^4 +
12 a^4 b^13 c x^2 y^2 z^4 - 18 a^12 b^4 c^2 x^2 y^2 z^4 -
48 a^11 b^5 c^2 x^2 y^2 z^4 + 264 a^10 b^6 c^2 x^2 y^2 z^4 -
336 a^9 b^7 c^2 x^2 y^2 z^4 + 276 a^8 b^8 c^2 x^2 y^2 z^4 -
336 a^7 b^9 c^2 x^2 y^2 z^4 + 264 a^6 b^10 c^2 x^2 y^2 z^4 -
48 a^5 b^11 c^2 x^2 y^2 z^4 - 18 a^4 b^12 c^2 x^2 y^2 z^4 +
304 a^11 b^4 c^3 x^2 y^2 z^4 - 976 a^10 b^5 c^3 x^2 y^2 z^4 +
1104 a^9 b^6 c^3 x^2 y^2 z^4 - 432 a^8 b^7 c^3 x^2 y^2 z^4 -
432 a^7 b^8 c^3 x^2 y^2 z^4 + 1104 a^6 b^9 c^3 x^2 y^2 z^4 -
976 a^5 b^10 c^3 x^2 y^2 z^4 + 304 a^4 b^11 c^3 x^2 y^2 z^4 +
492 a^10 b^4 c^4 x^2 y^2 z^4 - 1592 a^9 b^5 c^4 x^2 y^2 z^4 -
876 a^8 b^6 c^4 x^2 y^2 z^4 + 3952 a^7 b^7 c^4 x^2 y^2 z^4 -
876 a^6 b^8 c^4 x^2 y^2 z^4 - 1592 a^5 b^9 c^4 x^2 y^2 z^4 +
492 a^4 b^10 c^4 x^2 y^2 z^4 - 664 a^9 b^4 c^5 x^2 y^2 z^4 +
1032 a^8 b^5 c^5 x^2 y^2 z^4 + 5776 a^7 b^6 c^5 x^2 y^2 z^4 +
5776 a^6 b^7 c^5 x^2 y^2 z^4 + 1032 a^5 b^8 c^5 x^2 y^2 z^4 -
664 a^4 b^9 c^5 x^2 y^2 z^4 - 916 a^8 b^4 c^6 x^2 y^2 z^4 +
1552 a^7 b^5 c^6 x^2 y^2 z^4 + 8072 a^6 b^6 c^6 x^2 y^2 z^4 +
1552 a^5 b^7 c^6 x^2 y^2 z^4 - 916 a^4 b^8 c^6 x^2 y^2 z^4 +
624 a^7 b^4 c^7 x^2 y^2 z^4 + 816 a^6 b^5 c^7 x^2 y^2 z^4 +
816 a^5 b^6 c^7 x^2 y^2 z^4 + 624 a^4 b^7 c^7 x^2 y^2 z^4 +
142 a^6 b^4 c^8 x^2 y^2 z^4 - 388 a^5 b^5 c^8 x^2 y^2 z^4 +
142 a^4 b^6 c^8 x^2 y^2 z^4 - 276 a^5 b^4 c^9 x^2 y^2 z^4 -
276 a^4 b^5 c^9 x^2 y^2 z^4 + 294 a^4 b^4 c^10 x^2 y^2 z^4 +
4 a^16 b^2 x y^3 z^4 - 24 a^15 b^3 x y^3 z^4 +
52 a^14 b^4 x y^3 z^4 - 32 a^13 b^5 x y^3 z^4 -
56 a^12 b^6 x y^3 z^4 + 112 a^11 b^7 x y^3 z^4 -
56 a^10 b^8 x y^3 z^4 - 32 a^9 b^9 x y^3 z^4 +
52 a^8 b^10 x y^3 z^4 - 24 a^7 b^11 x y^3 z^4 +
4 a^6 b^12 x y^3 z^4 - 8 a^15 b^2 c x y^3 z^4 +
24 a^14 b^3 c x y^3 z^4 + 32 a^13 b^4 c x y^3 z^4 -
224 a^12 b^5 c x y^3 z^4 + 336 a^11 b^6 c x y^3 z^4 -
112 a^10 b^7 c x y^3 z^4 - 224 a^9 b^8 c x y^3 z^4 +
288 a^8 b^9 c x y^3 z^4 - 136 a^7 b^10 c x y^3 z^4 +
24 a^6 b^11 c x y^3 z^4 + 20 a^14 b^2 c^2 x y^3 z^4 -
64 a^13 b^3 c^2 x y^3 z^4 - 16 a^12 b^4 c^2 x y^3 z^4 +
128 a^11 b^5 c^2 x y^3 z^4 + 184 a^10 b^6 c^2 x y^3 z^4 -
576 a^9 b^7 c^2 x y^3 z^4 + 368 a^8 b^8 c^2 x y^3 z^4 -
44 a^6 b^10 c^2 x y^3 z^4 - 48 a^13 b^2 c^3 x y^3 z^4 +
144 a^12 b^3 c^3 x y^3 z^4 + 432 a^11 b^4 c^3 x y^3 z^4 -
1232 a^10 b^5 c^3 x y^3 z^4 + 432 a^9 b^6 c^3 x y^3 z^4 +
624 a^8 b^7 c^3 x y^3 z^4 - 304 a^7 b^8 c^3 x y^3 z^4 -
48 a^6 b^9 c^3 x y^3 z^4 - 8 a^12 b^2 c^4 x y^3 z^4 +
208 a^11 b^3 c^4 x y^3 z^4 + 136 a^10 b^4 c^4 x y^3 z^4 -
1824 a^9 b^5 c^4 x y^3 z^4 + 904 a^8 b^6 c^4 x y^3 z^4 +
2128 a^7 b^7 c^4 x y^3 z^4 - 1544 a^6 b^8 c^4 x y^3 z^4 +
128 a^11 b^2 c^5 x y^3 z^4 - 512 a^10 b^3 c^5 x y^3 z^4 -
1536 a^9 b^4 c^5 x y^3 z^4 + 5376 a^8 b^5 c^5 x y^3 z^4 +
4480 a^7 b^6 c^5 x y^3 z^4 + 256 a^6 b^7 c^5 x y^3 z^4 -
72 a^10 b^2 c^6 x y^3 z^4 + 144 a^8 b^4 c^6 x y^3 z^4 +
3136 a^7 b^5 c^6 x y^3 z^4 + 1912 a^6 b^6 c^6 x y^3 z^4 -
80 a^9 b^2 c^7 x y^3 z^4 + 368 a^8 b^3 c^7 x y^3 z^4 +
1072 a^7 b^4 c^7 x y^3 z^4 - 208 a^6 b^5 c^7 x y^3 z^4 +
68 a^8 b^2 c^8 x y^3 z^4 - 120 a^7 b^3 c^8 x y^3 z^4 -
316 a^6 b^4 c^8 x y^3 z^4 + 8 a^7 b^2 c^9 x y^3 z^4 -
24 a^6 b^3 c^9 x y^3 z^4 - 12 a^6 b^2 c^10 x y^3 z^4 +
a^18 y^4 z^4 - 6 a^17 b y^4 z^4 + 13 a^16 b^2 y^4 z^4 -
8 a^15 b^3 y^4 z^4 - 14 a^14 b^4 y^4 z^4 + 28 a^13 b^5 y^4 z^4 -
14 a^12 b^6 y^4 z^4 - 8 a^11 b^7 y^4 z^4 + 13 a^10 b^8 y^4 z^4 -
6 a^9 b^9 y^4 z^4 + a^8 b^10 y^4 z^4 - 6 a^17 c y^4 z^4 +
26 a^16 b c y^4 z^4 - 24 a^15 b^2 c y^4 z^4 -
56 a^14 b^3 c y^4 z^4 + 140 a^13 b^4 c y^4 z^4 -
84 a^12 b^5 c y^4 z^4 - 56 a^11 b^6 c y^4 z^4 +
104 a^10 b^7 c y^4 z^4 - 54 a^9 b^8 c y^4 z^4 +
10 a^8 b^9 c y^4 z^4 + 13 a^16 c^2 y^4 z^4 - 24 a^15 b c^2 y^4 z^4 -
52 a^14 b^2 c^2 y^4 z^4 + 120 a^13 b^3 c^2 y^4 z^4 +
46 a^12 b^4 c^2 y^4 z^4 - 232 a^11 b^5 c^2 y^4 z^4 +
140 a^10 b^6 c^2 y^4 z^4 + 8 a^9 b^7 c^2 y^4 z^4 -
19 a^8 b^8 c^2 y^4 z^4 - 8 a^15 c^3 y^4 z^4 -
56 a^14 b c^3 y^4 z^4 + 120 a^13 b^2 c^3 y^4 z^4 +
360 a^12 b^3 c^3 y^4 z^4 - 728 a^11 b^4 c^3 y^4 z^4 -
168 a^10 b^5 c^3 y^4 z^4 + 872 a^9 b^6 c^3 y^4 z^4 -
392 a^8 b^7 c^3 y^4 z^4 - 14 a^14 c^4 y^4 z^4 +
140 a^13 b c^4 y^4 z^4 + 46 a^12 b^2 c^4 y^4 z^4 -
728 a^11 b^3 c^4 y^4 z^4 + 334 a^10 b^4 c^4 y^4 z^4 +
716 a^9 b^5 c^4 y^4 z^4 - 494 a^8 b^6 c^4 y^4 z^4 +
28 a^13 c^5 y^4 z^4 - 84 a^12 b c^5 y^4 z^4 -
232 a^11 b^2 c^5 y^4 z^4 - 168 a^10 b^3 c^5 y^4 z^4 +
716 a^9 b^4 c^5 y^4 z^4 + 1788 a^8 b^5 c^5 y^4 z^4 -
14 a^12 c^6 y^4 z^4 - 56 a^11 b c^6 y^4 z^4 +
140 a^10 b^2 c^6 y^4 z^4 + 872 a^9 b^3 c^6 y^4 z^4 -
494 a^8 b^4 c^6 y^4 z^4 - 8 a^11 c^7 y^4 z^4 +
104 a^10 b c^7 y^4 z^4 + 8 a^9 b^2 c^7 y^4 z^4 -
392 a^8 b^3 c^7 y^4 z^4 + 13 a^10 c^8 y^4 z^4 -
54 a^9 b c^8 y^4 z^4 - 19 a^8 b^2 c^8 y^4 z^4 - 6 a^9 c^9 y^4 z^4 +
10 a^8 b c^9 y^4 z^4 + a^8 c^10 y^4 z^4 + 16 a^9 b^8 c x^3 z^5 -
80 a^8 b^9 c x^3 z^5 + 144 a^7 b^10 c x^3 z^5 -
80 a^6 b^11 c x^3 z^5 - 80 a^5 b^12 c x^3 z^5 +
144 a^4 b^13 c x^3 z^5 - 80 a^3 b^14 c x^3 z^5 +
16 a^2 b^15 c x^3 z^5 + 48 a^8 b^8 c^2 x^3 z^5 -
256 a^7 b^9 c^2 x^3 z^5 + 496 a^6 b^10 c^2 x^3 z^5 -
384 a^5 b^11 c^2 x^3 z^5 + 16 a^4 b^12 c^2 x^3 z^5 +
128 a^3 b^13 c^2 x^3 z^5 - 48 a^2 b^14 c^2 x^3 z^5 -
528 a^7 b^8 c^3 x^3 z^5 + 1104 a^6 b^9 c^3 x^3 z^5 -
416 a^5 b^10 c^3 x^3 z^5 - 352 a^4 b^11 c^3 x^3 z^5 +
176 a^3 b^12 c^3 x^3 z^5 + 16 a^2 b^13 c^3 x^3 z^5 -
496 a^6 b^8 c^4 x^3 z^5 + 768 a^5 b^9 c^4 x^3 z^5 +
32 a^4 b^10 c^4 x^3 z^5 - 384 a^3 b^11 c^4 x^3 z^5 +
80 a^2 b^12 c^4 x^3 z^5 + 1136 a^5 b^8 c^5 x^3 z^5 +
208 a^4 b^9 c^5 x^3 z^5 + 16 a^3 b^10 c^5 x^3 z^5 -
80 a^2 b^11 c^5 x^3 z^5 - 48 a^4 b^8 c^6 x^3 z^5 +
256 a^3 b^9 c^6 x^3 z^5 - 16 a^2 b^10 c^6 x^3 z^5 -
112 a^3 b^8 c^7 x^3 z^5 + 48 a^2 b^9 c^7 x^3 z^5 -
16 a^2 b^8 c^8 x^3 z^5 + 48 a^11 b^6 c x^2 y z^5 -
240 a^10 b^7 c x^2 y z^5 + 432 a^9 b^8 c x^2 y z^5 -
240 a^8 b^9 c x^2 y z^5 - 240 a^7 b^10 c x^2 y z^5 +
432 a^6 b^11 c x^2 y z^5 - 240 a^5 b^12 c x^2 y z^5 +
48 a^4 b^13 c x^2 y z^5 + 48 a^10 b^6 c^2 x^2 y z^5 -
384 a^9 b^7 c^2 x^2 y z^5 + 1008 a^8 b^8 c^2 x^2 y z^5 -
1152 a^7 b^9 c^2 x^2 y z^5 + 528 a^6 b^10 c^2 x^2 y z^5 -
48 a^4 b^12 c^2 x^2 y z^5 - 272 a^9 b^6 c^3 x^2 y z^5 +
80 a^8 b^7 c^3 x^2 y z^5 + 352 a^7 b^8 c^3 x^2 y z^5 +
416 a^6 b^9 c^3 x^2 y z^5 - 848 a^5 b^10 c^3 x^2 y z^5 +
272 a^4 b^11 c^3 x^2 y z^5 - 1552 a^8 b^6 c^4 x^2 y z^5 +
2176 a^7 b^7 c^4 x^2 y z^5 + 352 a^6 b^8 c^4 x^2 y z^5 -
1024 a^5 b^9 c^4 x^2 y z^5 + 48 a^4 b^10 c^4 x^2 y z^5 +
400 a^7 b^6 c^5 x^2 y z^5 + 2480 a^6 b^7 c^5 x^2 y z^5 +
1520 a^5 b^8 c^5 x^2 y z^5 - 560 a^4 b^9 c^5 x^2 y z^5 +
1424 a^6 b^6 c^6 x^2 y z^5 + 768 a^5 b^7 c^6 x^2 y z^5 -
80 a^4 b^8 c^6 x^2 y z^5 - 176 a^5 b^6 c^7 x^2 y z^5 +
240 a^4 b^7 c^7 x^2 y z^5 + 80 a^4 b^6 c^8 x^2 y z^5 +
48 a^13 b^4 c x y^2 z^5 - 240 a^12 b^5 c x y^2 z^5 +
432 a^11 b^6 c x y^2 z^5 - 240 a^10 b^7 c x y^2 z^5 -
240 a^9 b^8 c x y^2 z^5 + 432 a^8 b^9 c x y^2 z^5 -
240 a^7 b^10 c x y^2 z^5 + 48 a^6 b^11 c x y^2 z^5 -
48 a^12 b^4 c^2 x y^2 z^5 + 528 a^10 b^6 c^2 x y^2 z^5 -
1152 a^9 b^7 c^2 x y^2 z^5 + 1008 a^8 b^8 c^2 x y^2 z^5 -
384 a^7 b^9 c^2 x y^2 z^5 + 48 a^6 b^10 c^2 x y^2 z^5 +
272 a^11 b^4 c^3 x y^2 z^5 - 848 a^10 b^5 c^3 x y^2 z^5 +
416 a^9 b^6 c^3 x y^2 z^5 + 352 a^8 b^7 c^3 x y^2 z^5 +
80 a^7 b^8 c^3 x y^2 z^5 - 272 a^6 b^9 c^3 x y^2 z^5 +
48 a^10 b^4 c^4 x y^2 z^5 - 1024 a^9 b^5 c^4 x y^2 z^5 +
352 a^8 b^6 c^4 x y^2 z^5 + 2176 a^7 b^7 c^4 x y^2 z^5 -
1552 a^6 b^8 c^4 x y^2 z^5 - 560 a^9 b^4 c^5 x y^2 z^5 +
1520 a^8 b^5 c^5 x y^2 z^5 + 2480 a^7 b^6 c^5 x y^2 z^5 +
400 a^6 b^7 c^5 x y^2 z^5 - 80 a^8 b^4 c^6 x y^2 z^5 +
768 a^7 b^5 c^6 x y^2 z^5 + 1424 a^6 b^6 c^6 x y^2 z^5 +
240 a^7 b^4 c^7 x y^2 z^5 - 176 a^6 b^5 c^7 x y^2 z^5 +
80 a^6 b^4 c^8 x y^2 z^5 + 16 a^15 b^2 c y^3 z^5 -
80 a^14 b^3 c y^3 z^5 + 144 a^13 b^4 c y^3 z^5 -
80 a^12 b^5 c y^3 z^5 - 80 a^11 b^6 c y^3 z^5 +
144 a^10 b^7 c y^3 z^5 - 80 a^9 b^8 c y^3 z^5 +
16 a^8 b^9 c y^3 z^5 - 48 a^14 b^2 c^2 y^3 z^5 +
128 a^13 b^3 c^2 y^3 z^5 + 16 a^12 b^4 c^2 y^3 z^5 -
384 a^11 b^5 c^2 y^3 z^5 + 496 a^10 b^6 c^2 y^3 z^5 -
256 a^9 b^7 c^2 y^3 z^5 + 48 a^8 b^8 c^2 y^3 z^5 +
16 a^13 b^2 c^3 y^3 z^5 + 176 a^12 b^3 c^3 y^3 z^5 -
352 a^11 b^4 c^3 y^3 z^5 - 416 a^10 b^5 c^3 y^3 z^5 +
1104 a^9 b^6 c^3 y^3 z^5 - 528 a^8 b^7 c^3 y^3 z^5 +
80 a^12 b^2 c^4 y^3 z^5 - 384 a^11 b^3 c^4 y^3 z^5 +
32 a^10 b^4 c^4 y^3 z^5 + 768 a^9 b^5 c^4 y^3 z^5 -
496 a^8 b^6 c^4 y^3 z^5 - 80 a^11 b^2 c^5 y^3 z^5 +
16 a^10 b^3 c^5 y^3 z^5 + 208 a^9 b^4 c^5 y^3 z^5 +
1136 a^8 b^5 c^5 y^3 z^5 - 16 a^10 b^2 c^6 y^3 z^5 +
256 a^9 b^3 c^6 y^3 z^5 - 48 a^8 b^4 c^6 y^3 z^5 +
48 a^9 b^2 c^7 y^3 z^5 - 112 a^8 b^3 c^7 y^3 z^5 -
16 a^8 b^2 c^8 y^3 z^5 + 64 a^8 b^8 c^2 x^2 z^6 -
256 a^7 b^9 c^2 x^2 z^6 + 384 a^6 b^10 c^2 x^2 z^6 -
256 a^5 b^11 c^2 x^2 z^6 + 64 a^4 b^12 c^2 x^2 z^6 -
256 a^7 b^8 c^3 x^2 z^6 + 512 a^6 b^9 c^3 x^2 z^6 -
256 a^5 b^10 c^3 x^2 z^6 - 128 a^6 b^8 c^4 x^2 z^6 +
256 a^5 b^9 c^4 x^2 z^6 - 128 a^4 b^10 c^4 x^2 z^6 +
256 a^5 b^8 c^5 x^2 z^6 + 64 a^4 b^8 c^6 x^2 z^6 +
128 a^10 b^6 c^2 x y z^6 - 512 a^9 b^7 c^2 x y z^6 +
768 a^8 b^8 c^2 x y z^6 - 512 a^7 b^9 c^2 x y z^6 +
128 a^6 b^10 c^2 x y z^6 - 256 a^9 b^6 c^3 x y z^6 +
256 a^8 b^7 c^3 x y z^6 + 256 a^7 b^8 c^3 x y z^6 -
256 a^6 b^9 c^3 x y z^6 - 512 a^8 b^6 c^4 x y z^6 +
1024 a^7 b^7 c^4 x y z^6 - 512 a^6 b^8 c^4 x y z^6 +
256 a^7 b^6 c^5 x y z^6 + 256 a^6 b^7 c^5 x y z^6 +
384 a^6 b^6 c^6 x y z^6 + 64 a^12 b^4 c^2 y^2 z^6 -
256 a^11 b^5 c^2 y^2 z^6 + 384 a^10 b^6 c^2 y^2 z^6 -
256 a^9 b^7 c^2 y^2 z^6 + 64 a^8 b^8 c^2 y^2 z^6 -
256 a^10 b^5 c^3 y^2 z^6 + 512 a^9 b^6 c^3 y^2 z^6 -
256 a^8 b^7 c^3 y^2 z^6 - 128 a^10 b^4 c^4 y^2 z^6 +
256 a^9 b^5 c^4 y^2 z^6 - 128 a^8 b^6 c^4 y^2 z^6 +
256 a^8 b^5 c^5 y^2 z^6 + 64 a^8 b^4 c^6 y^2 z^6

Points on the Locus:

The circumcenter O is on the locus.
The barycentrics of the perspector are:

{a (a^4 - 2 a^2 b^2 + b^4 - a^2 b c + b^3 c - 2 a^2 c^2 - 4 b^2 c^2 + b c^3 + c^4 + 2 a b S - 2 b^2 S + 2 a c S + 4 b c S - 2 c^2 S),
b (a^4 - 2 a^2 b^2 + b^4 + a^3 c - a b^2 c - 4 a^2 c^2 - 2 b^2 c^2 + a c^3 + c^4 - 2 a^2 S + 2 a b S + 4 a c S + 2 b c S - 2 c^2 S),
c (a^4 + a^3 b - 4 a^2 b^2 + a b^3 + b^4 - 2 a^2 c^2 - a b c^2 - 2 b^2 c^2 + c^4 - 2 a^2 S + 4 a b S - 2 b^2 S + 2 a c S +
2 b c S)}

where S is twice the area of ABC, not in ETC, perhaps a simpler form is possible.

ΠΡΟΣΘΗΚΗ 5/9/19
The point with its "brother" are now in ETC
X(34215)= X(4)X(175)∩X(8)X(492)
X(34216)= X(4)X(176)∩X(8)X(491)

INRADIUS 3

Let ABC be a triangle D a point on BC and r_1,r_2 the inradii of ABD, ACD resp. To construct ABC if we know the angles of ABC and are given the r_1,r_2.

Solution:

Let AE = h_a be the altitude from A and r the inradius of ABC.

We have:

h_a = 2r_1r_2 / (r_1 + r_2 - r)

(by this Theorem)

and

r / h_a = 4Rsin(A/2)sin(B/2)sin(C/2)/2RsinBsinC = sin(A/2)/2cos(B/2)cos(C/2)

==> h_a is known.

Addendum (12-1-2011):

Synthetic solution by Nikolaos Dergiades:
Hyacinthos, Message 19725

NO COMPUTATIONS, PLEASE! -- 1

Problem:

Let ABC be a triangle, P a point, A'B'C' the pedal triangle of P and Ab, Ac the orthogonal projections of A' on PB',PC', resp. The line AbAc intersects BC at A". Similarly B' and C'. Which is the locus of P such that the points A',B',C' are collinear?

Lemma:

Let BC be a line segment, A' a fixed point on BC and A a variable point on the perpendicular to BC at A'. Let Ab,Ac be the orthogonal projections of A' on AB,AC, resp. The line AbAc passes through a fixed point.

Let A" be the intersection of the lines BC and AbAc.

We have that:

A"B/A"C = (A'B/A'C)^2 (the proof is left to the reader).

Therefore AbAc passes through the fixed point A".

Now, in the Problem we have:

A"B/A"C = (A'B/A'C)^2

and similarly:

B"C/C"A = (B'C/B'A)^2

C"A/A"B = (C'A/C'B)^2

A",B",C" are collinear ==>

(A"B/A"C).(B"C/C"A).(C"A/A"B) = [(A'B/A'C).(B'C/B'A).(C'A/C'B)]^2 = 1

==>

1. (A'B/A'C).(B'C/B'A).(C'A/C'B) = -1 ==> ABC, A'B'C' are perspective
(ie AA',BB',CC' are concurrent, by Ceva Theorem)

Or

2. (A'B/A'C).(B'C/B'A).(C'A/C'B) = +1 ==> A',B',C' are collinear (by Menelaus Theorem).

1: The locus is the Darboux cubic
2: The locus is the Circumcircle + the Line at Infinity.

Therefore the locus of P is the union of the Darboux cubic, the Circumcircle, and the Line at Infinity.