PARALLEL NN-LINES

10. Let ABC be a triangle and P a point.

Denote:

Ab, Ac = the orthogonal projections of A on PB,PC, resp.

Na1 = the NPC center of AAbAc

Na2 = the NPC center of Na1AbAc.

Similarly Nb1, Nb2 and Nc1, Nc2.

The lines Na1Na2, Nb1Nb2, Nc1Nc2 are parallel.

11. If P = I,

the lines Na1Na2, Nb1Nb2, Nc1Nc2 are parallel to Euler Line of ABC

21. Let ABC be a triangle.

Denote:

Na1 = the NPC center of IBC

Na2 = the NPC center of Na1BC.

Similarly Nb1,Nb2, Nc1,Nc2

The lines Na1Na2, Nb1Nb2, Nc1Nc2 are parallel to Euler Line of ABC

31. Let ABC be a triangle and IaIbIc the antipedal triangle of I (excentral triangle)

Denote:

Ab, Ac = the orthogonal projections of A on IaIc, IaIb, resp.

Na1 = the NPC center of AAbAc

Na2 = the NPC center of Na1AbAc

The lines Na1Na2, Nb1Nb2, Nc1Nc2 are parallel to Euler Line of ABC

41. Let ABC be a triangle and IaIbIc the antipedal triangle of I (excentral triangle)

Denote:

Na1 = the NPC center of IaBC

Oa = the circumcenter of IaBC

Nao1 = The NPC center of OaBC.

Similarly Nb1, Nbo1, Nc1, Nco1.

The lines Na1Nao1, Nb1Nbo1, Nc1Nco1 are parallel to OI line of ABC.

Antreas P. Hatzipolakis, 1 October 2014

Canarina canariensis

Re: RADICAL CENTERS - NPC - OI LINE

Posted By: amontes1949

Wed Aug 21, 2013 4:44 am

[Antreas P. Hatzipolakis]:

Let ABC be a triangle and A'B'C' the cevian triangle of I.

Denote:

(Nab), (Nac) = the NPCs of AIB', AIC', resp.

(Nbc), (Nba) = the NPCs of BIC', BIA', resp.

(Nca), (Ncb) = the NPCs of CIA', CIB', resp.

R = the radical center of (Nbc), (Nca), (Nab)

S = the radical center of (Nba), (Ncb), (Nac)

*** (Trilinear ccordinates)

R = ((a + b - c) (a - b + c) (a^3 b - a b^3 + 2 a^3 c - 2 a b^2 c - b^3 c + a^2 c^2 - 3 a b c^2 - 3 b^2 c^2 - 2 a c^3 - 3 b c^3 - c^4) : (a - b - c) (a + b - c) (a^4 + 2 a^3 b - a^2 b^2 - 2 a b^3 + 3 a^3 c + 3 a^2 b c - b^3 c + 3 a^2 c^2 + 2 a b c^2 + a c^3 + b c^3) : (a - b - c) (a - b + c) (a^3 b + 3 a^2 b^2 + 3 a b^3 + b^4 + a^3 c + 2 a^2 b c + 3 a b^2 c + 2 b^3 c - b^2 c^2 - a c^3 - 2 b c^3)) T = ((a + b - c) (a - b + c) (2 a^3 b + a^2 b^2 - 2 a b^3 - b^4 + a^3 c - 3 a b^2 c - 3 b^3 c - 2 a b c^2 - 3 b^2 c^2 - a c^3 - b c^3): (a - b - c) (a + b - c) (a^3 b - a b^3 + a^3 c + 2 a^2 b c - 2 b^3 c + 3 a^2 c^2 + 3 a b c^2 - b^2 c^2 + 3 a c^3 + 2 b c^3 + c^4) : (a - b - c) (a - b + c) (a^4 + 3 a^3 b + 3 a^2 b^2 + a b^3 + 2 a^3 c + 3 a^2 b c + 2 a b^2 c + b^3 c - a^2 c^2 - 2 a c^3 - b c^3) )

R and S is a bicentric pair, then the midpoint of RS is a triangle center of trilinear ccordinates:

M = ( (a + b - c) (a - b + c) (3 a^3 b + a^2 b^2 - 3 a b^3 - b^4 + 3 a^3 c - 5 a b^2 c - 4 b^3 c + a^2 c^2 - 5 a b c^2 - 6 b^2 c^2 - 3 a c^3 - 4 b c^3 - c^4) : (a - b - c) (a + b - c) (a^4 + 3 a^3 b - a^2 b^2 - 3 a b^3 + 4 a^3 c + 5 a^2 b c - 3 b^3 c + 6 a^2 c^2 + 5 a b c^2 - b^2 c^2 + 4 a c^3 + 3 b c^3 + c^4) : (a - b - c) (a - b + c) (a^4 + 4 a^3 b + 6 a^2 b^2 + 4 a b^3 + b^4 + 3 a^3 c + 5 a^2 b c + 5 a b^2 c + 3 b^3 c - a^2 c^2 - b^2 c^2 - 3 a c^3 - 3 b c^3) )

M lie on the central line X(1)X(3).

Ideal point of line RT is X(513).

This bicentric pair does not appear in the current edition of "BICENTRIC PAIRS OF POINTS" ( Clark Kimberling)

I suggest as a flower name for this bicentric pair: Canarina

Angel Montesdeoca

Anopolis #860

NPCs. SEQUENCE OF POINTS

Let ABC be a triangle and P a point.

Denote:

A1, B1, C1 = The NPC centers of PBC, PCA, PAB, resp.

r11,r12,r13 = the radical axes of the NPCs: ((B1),(C1)), ((C1),(A1)), ((A1),(B1)), resp.

R1 = the point of concurrence of r11,r12,r13 [Radical center of the circles. It is the Poncelet point of P wrt ABC, lying on the NPC of ABC]

f11, f12, f13 = the parallels to r11, r12, r13 through A,B,C, resp.

The lines f11, f12, f13 concur at a point F1.

F1 is the reflection of P in R1.

Locus:

As P moves on a line (the Euler line, for example) which is the locus of F1 ?

Sequences of Ri, Fi:

Denote:

A2,B2,C2 = the NPC centers of A1BC, B1CA, C1AB, resp.

r21,r22,r23 = the radical axes of the NPC: ((B2),(C2)), ((C2),(A2)), ((A2),(B2)), resp.

R2 = the point of concurrence of r21,r22,r23

f21, f22, f23 = the parallels to r21, r22, r23 through A,B,C, resp.

The lines f21, f22, f23 concur at a point F2.

Similarly R3,R4...., Rn and F3,F4,.... Fn.

Where are lying the points Ri, Fi ?

Special points P: P = N, I

Antreas P. Hatzipolakis, 9 May 2013

CONCYCLIC POINTS. LOCUS

A CIRCLE:

Let ABC be a triangle, A'B'C' the cevian triangle of I and N1, N2, N3 the NPC centers of IB'C', IC'A', IA'B', resp.

The points I, N1,N2,N3 are concyclic.

ETC X(5453)

Center of the circle?

LOCUS:

Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P and N1, N2, N3 the NPC centers of PB'C', PC'A', PA'B', resp.

Which is the locus of P such that the points P,N1,N2,N3 are concyclic ?

Antreas P. Hatzipolakis, 17 April 2013, Hyacinthos #21970

**********************************************************

The center X of the circle has trilinears:

2*cos(A)+4*sin(3*A/2)*cos(B/2-C/2)+ cos(B-C)+2 : :

ETC search: 2.387773069046934.., 2.38593313995937.., 0.886815506990847..

X = Midpoint of X(I),X(J) for these (I,J): (1,500)

X lies on line X(I),X(J) for these (I,J):

(1,30), (3,81), (5,581), (21,323), (58,5428), (140,3216), (186,2906), (386,549), (511,1385), (550,991), (1154,2646), (2771,3743)

Cιsar Lozada Hyacinthos #21972

**********************************************************

A related locus:

Q such that Q and the NPCs of BCQ, CAQ, ABQ are concyclic. This would include X(13), X(14), the bicentric pair PU(5), the circumcircle intercepts of line X(5)X(523), and the point Qi (of ABC) such that I = Qi of the cevian triangle of I.

Randy Hutson, Hyacinthos #21977

This is the tricircular sextic: Q014 - 4 S^2 x y z (x + y + z) (c^2 x y + b^2 x z + a^2 y z) where S = twice the area of ABC. (tricircular = the circular points are triple points of the curve).

Francisco Javier Hyacinthos #21981

.... the point Qi on this curve is the point which is the incenter of its anticevian triangle, and has ETC search value 1.999434154060428. Coordinates? Its isogonal conjugate Qi* is the point which is the nine-point center of its pedal triangle (ETC search 1.142779079509848). The line QiQi* passes through X(5).

Randy Hutson, Hyacinthos #21982

Does this locus contain any ETC centers or bicentric pairs besides X(13), X(14), and PU(5)? Are the circumcircle intercepts of line X(5)X(523) triangle centers or another bicentric pair?

Randy Hutson, Hyacinthos #21983

QUINTIC

Let ABC be a triangle, P a point and A1,B1,C1 are the NPC centers of PBC, PCA, PAB, resp.

Which is the locus of P such that the circumcenter O0 of A1B1C1 lies on the Euler line?

P : X1 = I, X4= H, X5 = N.......

For P = I we have O0 = N

Antreas P. Hatzipolakis, 15 April 2013

****************************************

It is a circular quintic through I, H and N. The other three intersection points with Euler line are X30 (the infinite point) and the instersections with the circumcircle X1113 and X1114.

Equation:

a^6 c^2 x^3 y^2 - 3 a^4 b^2 c^2 x^3 y^2 + 3 a^2 b^4 c^2 x^3 y^2 - b^6 c^2 x^3 y^2 - a^4 c^4 x^3 y^2 + a^2 b^2 c^4 x^3 y^2 - a^2 c^6 x^3 y^2 + c^8 x^3 y^2 + a^6 c^2 x^2 y^3 - 3 a^4 b^2 c^2 x^2 y^3 + 3 a^2 b^4 c^2 x^2 y^3 - b^6 c^2 x^2 y^3 - a^2 b^2 c^4 x^2 y^3 + b^4 c^4 x^2 y^3 + b^2 c^6 x^2 y^3 - c^8 x^2 y^3 + 3 a^2 b^4 c^2 x^3 y z - 3 b^6 c^2 x^3 y z - 3 a^2 b^2 c^4 x^3 y z + 3 b^2 c^6 x^3 y z + 2 a^6 c^2 x^2 y^2 z - 2 b^6 c^2 x^2 y^2 z - 4 a^4 c^4 x^2 y^2 z + 4 b^4 c^4 x^2 y^2 z + 2 a^2 c^6 x^2 y^2 z - 2 b^2 c^6 x^2 y^2 z + 3 a^6 c^2 x y^3 z - 3 a^4 b^2 c^2 x y^3 z + 3 a^2 b^2 c^4 x y^3 z - 3 a^2 c^6 x y^3 z - a^6 b^2 x^3 z^2 + a^4 b^4 x^3 z^2 + a^2 b^6 x^3 z^2 - b^8 x^3 z^2 + 3 a^4 b^2 c^2 x^3 z^2 - a^2 b^4 c^2 x^3 z^2 - 3 a^2 b^2 c^4 x^3 z^2 + b^2 c^6 x^3 z^2 - 2 a^6 b^2 x^2 y z^2 + 4 a^4 b^4 x^2 y z^2 - 2 a^2 b^6 x^2 y z^2 + 2 b^6 c^2 x^2 y z^2 - 4 b^4 c^4 x^2 y z^2 + 2 b^2 c^6 x^2 y z^2 + 2 a^6 b^2 x y^2 z^2 - 4 a^4 b^4 x y^2 z^2 + 2 a^2 b^6 x y^2 z^2 - 2 a^6 c^2 x y^2 z^2 + 4 a^4 c^4 x y^2 z^2 - 2 a^2 c^6 x y^2 z^2 + a^8 y^3 z^2 - a^6 b^2 y^3 z^2 - a^4 b^4 y^3 z^2 + a^2 b^6 y^3 z^2 + a^4 b^2 c^2 y^3 z^2 - 3 a^2 b^4 c^2 y^3 z^2 + 3 a^2 b^2 c^4 y^3 z^2 - a^2 c^6 y^3 z^2 - a^6 b^2 x^2 z^3 + b^8 x^2 z^3 + 3 a^4 b^2 c^2 x^2 z^3 + a^2 b^4 c^2 x^2 z^3 - b^6 c^2 x^2 z^3 - 3 a^2 b^2 c^4 x^2 z^3 - b^4 c^4 x^2 z^3 + b^2 c^6 x^2 z^3 - 3 a^6 b^2 x y z^3 + 3 a^2 b^6 x y z^3 + 3 a^4 b^2 c^2 x y z^3 - 3 a^2 b^4 c^2 x y z^3 - a^8 y^2 z^3 + a^2 b^6 y^2 z^3 + a^6 c^2 y^2 z^3 - a^4 b^2 c^2 y^2 z^3 - 3 a^2 b^4 c^2 y^2 z^3 + a^4 c^4 y^2 z^3 + 3 a^2 b^2 c^4 y^2 z^3 - a^2 c^6 y^2 z^3 = 0

Francisco Javier, Hyacinthos #21962

SEQUENCE OF POINTS ON THE EULER LINE

Let ABC be a triangle.

Denote:

A1, B1, C1 = The NPC centers of NBC, NCA, NAB, resp.

A2, B2 ,C2 = The NPC centers of A1BC, B1CA, C1AB, resp.

A3, B3, C3 = The NPC centers of A2BC, B2CA, C2AB

An, Bn, Cn = The NPC centers of A_n-1BC, B_n-1CA, C_n-1AB.

On = the circumcenter of the triangle AnBnCn.

The points O1, O2,......, On, ..... lie on the Euler Line of ABC.

The point O1 is now in ETC: X5501

Coordinates of On? Ratio of OnO / OnN ?

Antreas P. Hatzipolakis, 14 April 2013

********************************************

I calculated the ratios NO1:O1O and NO2:O2O. The expression for the first one is quite long, and that for the second one is enormous:

If p stands for the semiperimeter, we have

NO1:O1O = (2 p^4 - 12 p^2 r^2 + 2 r^4 - 16 p^2 r R + 16 r^3 R - 8 p^2 R^2 + 40 r^2 R^2 + 32 r R^3 + 9 R^4)/(2 p^4 + 20 p^2 r^2 + 2 r^4 - 16 p^2 r R + 16 r^3 R - 16 p^2 R^2 + 48 r^2 R^2 + 64 r R^3 + 23 R^4)

Francisco Javier, Hyacinthos #21952

The conjecture is false !!

From Cesar Lozada (24 June 2016)

Dear Antreas,

Sorry. O1 does lie on Euler line but O2 does not. Algebraically confirmed.

I didn´t try O3 because calculus are awful

Regards,

César Lozada

CIRCUMCENTER OF N1N2N3

Let ABC be a triangle, P a point and N1,N2,N3 the NPC centers of PBC, PCA, PAB, resp.

Which is the circumcenter Op of N1N2N3 ?

P = H. Op = N [N1 = N2 = N3 = N (N1N2N3 is degenerated)]

P = I. Op = N

P = O. Op lies on the line NU, where U is the Poncelet point of O wrt ABC (ie the point where concur the NPCs of OBC,OCA,OAB and ABC)

P = N. Op lies on the Euler line of ABC.

If P describes a line (Euler line, etc) which is the locus of Op?

If P describes the circumcircle, Op is the Poncelet point of P wrt ABC. If P describes other curves?

Antreas P. Hatzipolakis, 14 April 2013

********************************************************

For P = O, Op = complement of X(157).

In general, Op = complement of the nine-point-center [N'] of the antipedal triangle [A'B'C'] of P.

Randy Hutson, Hyacinthos #21963

INCENTER AND NPCs

Let ABC be a triangle.

Denote:

N1,N2,N3 = The NPC centers of IBC, ICA, IAB, resp.

N11, N22, N33 = The NPC centers of N1BC, N2CA, N3AB, resp.

1. The lines N1N11, N2N22, N3N33 are parallel to Euler Line of ABC.

2. The Circumcenter of N1N2N3 is N, the NPC center of ABC.

Synthetic Proofs?

Antreas P. Hatzipolakis, 13 April 2013

RADICAL AXES OF NPCs

Let ABC be a triangle and P,P* two isogonal conjugate points.

Denote:

Ra = the radical axis of the circles (NPC of PBC), (NPC of P*BC)

Rb = the radical axis of the circles (NPC of PCA), (NPC of P*CA)

Rc = the radical axis of the circles (NPC of PAB), (NPC of P*AB)

Ra, Rb, Rc are concurrent.

Antreas P. Hatzipolakis, 12 April 2013

ORTHOLOGIC, EULER LINE

Let ABC be a triangle, A'B'C', A"B"C" the medial, orthic triangles, resp. and A*,B*,C* points on AA",BB",CC", resp. such that: A*A/A*A" = B*B/B*B" = C*C/C*C" = t.

Denote:

Ab = (Parallel to BC through A*) /\ AC

Ac = (Parallel to BC through A*) /\ AB

Bc = (Parallel to CA through B*) /\ BA

Ba = (Parallel to CA through B*) /\ BC

Ca = (Parallel to AB through C*) /\ CB

Cb = (Parallel to AB through C*) /\ CA

1. Oa, Ob, Oc = the circumcenters of A'AbAc, B'BcBa, C'CaCb, resp.

The triangles ABC, OaObOc are orthologic

The locus of the orthologic center (OaObOc, ABC), as t varies, is the Euler line of ABC.

Which is the locus of the other orthologic center (ABC, OaObOc)?

2. Na, Nb, Nc = the NPCs centers of A'AbAc, B'BcBa, C'CaCb, resp.

The triangles ABC, NaNbNc are orthologic

The locus of the orthologic center (NaNbNc, ABC), as t varies, is the Euler line of ABC.

Which is the locus of the other orthologic center (ABC, NaNbNc)?

Antreas P. Hatzipolakis, 6 April 2013

ORTHOPOLAR CIRCLES

Let ABCD be a quadrilateral [quadragon], A',B',C',D' the circumcenters of BCD, CDA, DAB, ABC, resp. and P a point.

Denote:

1 = the orthopole of PA' wrt BCD

2 = the orthopole of PB' wrt CDA

3 = the orthopole of PC' wrt DAB

4 = the orthopole of PD' wrt ABC

Conjecture:

The points 1,2,3,4 are concyclic. The circle passes through the Poncelet point of ABCD (=the point where the NPCs of BCD, CDA, DAB, ABC concur)

The circle (1,2,3,4) is a line when ABCD is cyclic (ie A' = B' = C'= D' = O ==> PO = PA' = PB' = PC' := L, a line psiing through the circumcenter of the cyclic ABCD)

Antreas P. Hatzipolakis, 30 March 2013

ORTHOPOLAR CIRCLES [triangle]

Let ABC be a triangle, P, Q two points and O, Q1,Q2,Q3 the circumcenters of ABC, QBC, QCA, QAB, resp.

Denote:

P0 = the orthopole of PO wrt ABC

P1 = the orthopole of PQ1 wrt QBC

P2 = the orthopole of PQ2 wrt QCA

P3 = the orthopole of PQ3 wrt QAB

We have:

1. P0, P1, P2, P3 lie on the NPCs (N),(N1),(N2),(N3) of ABC, QBC, QCA, QAB, resp. (since the respective lines pass through the circumcenters of the respective triangles)

2. The NPCs of ABC, QBC, QCA, QAB concur at the Poncelet point Q* of Q wrt ABC.

CONJECTURE:

The points P0, P1, P2, P3, Q* are concyclic.

Antreas P. Hatzipolakis, 30 March 2013.

REFLECTING NPCs

Let ABC be a triangle.

1. Let (N1),(N2),(N3) be the reflections of the NPC (N) in the sidelines BC,CA,AB, resp. and A'B'C' the triangle bounded by the radical axes of ((O),(N1)), ((O),(N2)), ((O),(N3)), resp.
The triangles ABC, A'B'C' are perspective.

2. Let HaHbHc be the orthic triangle, (N1),(N2),(N3) the reflections of the NPC (N) in the altitudes HHa,HHb,HHc, resp. and and A'B'C' the triangle bounded by the radical axes of ((O),(N1)), ((O),(N2)), ((O),(N3)), resp.
The triangles HaHbHc, A'B'C' are perspective.

3. Let OaObOc be the medial triangle [pedal tr. of O], (N1),(N2),(N3) the reflections of the NPC (N) in the perp. bisectors OOa,OOb,OOc, resp. and and A'B'C' the triangle bounded by the radical axes of ((O),(N1)), ((O),(N2)), ((O),(N3)), resp.
The triangles ABC, A'B'C' are perspective.

Antreas P. Hatzipolakis, 11 Febr. 2013

COLLINEAR NPC CENTERS ?

Let ABC be a triangle, L a line passing through H (orthocenter), intersecting the sidelines BC,CA,AB at A',B',C', resp., La,Lb,Lc the reflections of L in the sidelines BC,CA,AB, resp. (concurrent at a point S on the circumcircle) and Ab,Ac the orthogonal projections of A' on Lb,Lc, resp., Bc,Ba the orthogonal projections of B' on Lc,La, resp. and Ca,Cb the orthogonal projections of C' on La,Lb, resp. The NPC centers Na,Nb,Nc of of A'AbAc, B'BcBa, C'CaCb resp. are collinear. (??)

Antreas P. Hatzipolakis, 30 Jan. 2013

PICASSO POINT [X(4550]

Let ABC be a triangle, A1B1C1 the pedal triangle of H, A2B2C2 the antipodal triangle of A1B1C1 with respect the circumcircle of A1B1C1 (= NPC of ABC), A'B'C' the antipodal triangle of ABC (= circumcevian triangle of O).

Denote:

A" = The other than A' intersection of A'A2 and the circumcircle of ABC

B" = The other than B' intersection of B'B2 and the circumcircle of ABC

C" = The other than C' intersection of C'C2 and the circumcircle of ABC

N1 = The NPC Center of A"BC

N2 = The NPC center of B"CA

N3 = The NPC center of C"AB

The four NPC centers N, N1,N2,N3 are concyclic (??).

Center of the circle ?

APH
14 January 2012

---------------------------------------------

Yes. The center is the point:

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


The radius of the circle has a long expression:

-(1/(4 (p^2 - r^2 - 4 r R)^2))(3 p^6 - 5 p^4 r^2 + 5 p^2 r^4 -
3 r^6 - 36 p^4 r R + 56 p^2 r^3 R - 36 r^5 R - 46 p^4 R^2 +
224 p^2 r^2 R^2 - 190 r^4 R^2 + 368 p^2 r R^3 - 560 r^3 R^3 +
216 p^2 R^4 - 952 r^2 R^4 - 864 r R^5 - 324 R^6)

where p= semiperimeter.

Francisco Javier García Capitán
15 January 2012

Now in ETC X(4550)
--------------------------------------------

I name this new point as PICASSO Point in honor of great Pablo Picasso, whose a painting was recently stolen from the Greek National Gallery. See HERE and also my article in my blog Madara HERE

Antreas

--------------------------------------------

This point P lies on line G-X74 and satisfies the ratio

GP:PX74= (2 p^2 - 2 r^2 - 8 r R - 9 R^2)/(9 R^2).

Francisco, Hyacinthos 20683

Like Segovia point, also lies in line H-X1209, moreover Picasso point is the midpoint of H and Segovia point.

Francisco, Hyacinthos 20684

--------------------------------------------

The four nine-point centers are concyclic on a circle with center

(a^2(S^2+3S_{AA})(S^2+3S_{BC}) : ... : ...)

in barycentric coordinates.

This point lies on the line joining the centroid to X(74), the fourth intersection of the circumcircle with the Jerabek hyperbola.

Paul Yiu

--------------------------------------------

Denote:

Na = The NPC Center of AB"C"

Nb = The NPC center of BC"A"

Nc = The NPC center of CA"B"

The radical axes of the NPCs (N1) and (Na) - (N2) and (Nb) - (N3) and (Nc) are concurrent [See NPCs and Radical Axes], or equivalently the triangles N1N2N3, NaNbNc are perspective.

Antreas

NPCs and Radical Axes

Let ABC be a triangle, P a point, A1B1C1 and A2B2C2 its cevian and cyclocevian triangles, resp.

Denote:

R1 := the radical axis of the NPCs of A1B2C2, A2B1C1

R2 := the radical axis of the NPCs of B1C2A2, B2C1A1

R3 := the radical axis of the NPCs of C1A2B2, C2A1B1

The lines R1,R2,R3 are concurrent.
(and also the radical axis of the NPCs of A1B1C1,A2B2C2)

Point of concurrence?

Variation:

-- A'B'C' = Circumcevian triangle of point P wrt ABC.

Generalization:

-- A1B1C1, A2B2C2 = two triangles inscribed in the same circle.

APH, 14 December 2011

------------------------------------------------

For the triangle case [cyclocevian] this is the point:


{a^2 (-a^4 c^4 u^3 v^3 + b^4 c^4 u^3 v^3 + 2 a^2 c^6 u^3 v^3 -
c^8 u^3 v^3 + a^6 c^2 u^3 v^2 w + 2 a^4 b^2 c^2 u^3 v^2 w -
7 a^2 b^4 c^2 u^3 v^2 w + 4 b^6 c^2 u^3 v^2 w -
3 a^4 c^4 u^3 v^2 w - 6 a^2 b^2 c^4 u^3 v^2 w -
7 b^4 c^4 u^3 v^2 w + 3 a^2 c^6 u^3 v^2 w + 4 b^2 c^6 u^3 v^2 w -
c^8 u^3 v^2 w + a^6 c^2 u^2 v^3 w - 3 a^2 b^4 c^2 u^2 v^3 w +
2 b^6 c^2 u^2 v^3 w - 4 a^4 c^4 u^2 v^3 w -
6 a^2 b^2 c^4 u^2 v^3 w - 6 b^4 c^4 u^2 v^3 w +
5 a^2 c^6 u^2 v^3 w + 6 b^2 c^6 u^2 v^3 w - 2 c^8 u^2 v^3 w +
a^6 b^2 u^3 v w^2 - 3 a^4 b^4 u^3 v w^2 + 3 a^2 b^6 u^3 v w^2 -
b^8 u^3 v w^2 + 2 a^4 b^2 c^2 u^3 v w^2 -
6 a^2 b^4 c^2 u^3 v w^2 + 4 b^6 c^2 u^3 v w^2 -
7 a^2 b^2 c^4 u^3 v w^2 - 7 b^4 c^4 u^3 v w^2 +
4 b^2 c^6 u^3 v w^2 + 2 a^6 b^2 u^2 v^2 w^2 -
6 a^4 b^4 u^2 v^2 w^2 + 6 a^2 b^6 u^2 v^2 w^2 -
2 b^8 u^2 v^2 w^2 + 2 a^6 c^2 u^2 v^2 w^2 -
10 a^2 b^4 c^2 u^2 v^2 w^2 + 8 b^6 c^2 u^2 v^2 w^2 -
6 a^4 c^4 u^2 v^2 w^2 - 10 a^2 b^2 c^4 u^2 v^2 w^2 -
12 b^4 c^4 u^2 v^2 w^2 + 6 a^2 c^6 u^2 v^2 w^2 +
8 b^2 c^6 u^2 v^2 w^2 - 2 c^8 u^2 v^2 w^2 + a^6 b^2 u v^3 w^2 -
3 a^4 b^4 u v^3 w^2 + 3 a^2 b^6 u v^3 w^2 - b^8 u v^3 w^2 +
2 a^6 c^2 u v^3 w^2 - 2 a^4 b^2 c^2 u v^3 w^2 -
4 a^2 b^4 c^2 u v^3 w^2 + 4 b^6 c^2 u v^3 w^2 -
5 a^4 c^4 u v^3 w^2 - 3 a^2 b^2 c^4 u v^3 w^2 -
6 b^4 c^4 u v^3 w^2 + 4 a^2 c^6 u v^3 w^2 + 4 b^2 c^6 u v^3 w^2 -
c^8 u v^3 w^2 - a^4 b^4 u^3 w^3 + 2 a^2 b^6 u^3 w^3 -
b^8 u^3 w^3 + b^4 c^4 u^3 w^3 + a^6 b^2 u^2 v w^3 -
4 a^4 b^4 u^2 v w^3 + 5 a^2 b^6 u^2 v w^3 - 2 b^8 u^2 v w^3 -
6 a^2 b^4 c^2 u^2 v w^3 + 6 b^6 c^2 u^2 v w^3 -
3 a^2 b^2 c^4 u^2 v w^3 - 6 b^4 c^4 u^2 v w^3 +
2 b^2 c^6 u^2 v w^3 + 2 a^6 b^2 u v^2 w^3 - 5 a^4 b^4 u v^2 w^3 +
4 a^2 b^6 u v^2 w^3 - b^8 u v^2 w^3 + a^6 c^2 u v^2 w^3 -
2 a^4 b^2 c^2 u v^2 w^3 - 3 a^2 b^4 c^2 u v^2 w^3 +
4 b^6 c^2 u v^2 w^3 - 3 a^4 c^4 u v^2 w^3 -
4 a^2 b^2 c^4 u v^2 w^3 - 6 b^4 c^4 u v^2 w^3 +
3 a^2 c^6 u v^2 w^3 + 4 b^2 c^6 u v^2 w^3 - c^8 u v^2 w^3 +
a^6 b^2 v^3 w^3 - 2 a^4 b^4 v^3 w^3 + a^2 b^6 v^3 w^3 +
a^6 c^2 v^3 w^3 - a^2 b^4 c^2 v^3 w^3 - 2 a^4 c^4 v^3 w^3 -
a^2 b^2 c^4 v^3 w^3 + a^2 c^6 v^3 w^3),
b^2 (a^4 c^4 u^3 v^3 - b^4 c^4 u^3 v^3 + 2 b^2 c^6 u^3 v^3 -
c^8 u^3 v^3 + 2 a^6 c^2 u^3 v^2 w - 3 a^4 b^2 c^2 u^3 v^2 w +
b^6 c^2 u^3 v^2 w - 6 a^4 c^4 u^3 v^2 w -
6 a^2 b^2 c^4 u^3 v^2 w - 4 b^4 c^4 u^3 v^2 w +
6 a^2 c^6 u^3 v^2 w + 5 b^2 c^6 u^3 v^2 w - 2 c^8 u^3 v^2 w +
4 a^6 c^2 u^2 v^3 w - 7 a^4 b^2 c^2 u^2 v^3 w +
2 a^2 b^4 c^2 u^2 v^3 w + b^6 c^2 u^2 v^3 w -
7 a^4 c^4 u^2 v^3 w - 6 a^2 b^2 c^4 u^2 v^3 w -
3 b^4 c^4 u^2 v^3 w + 4 a^2 c^6 u^2 v^3 w + 3 b^2 c^6 u^2 v^3 w -
c^8 u^2 v^3 w - a^8 u^3 v w^2 + 3 a^6 b^2 u^3 v w^2 -
3 a^4 b^4 u^3 v w^2 + a^2 b^6 u^3 v w^2 + 4 a^6 c^2 u^3 v w^2 -
4 a^4 b^2 c^2 u^3 v w^2 - 2 a^2 b^4 c^2 u^3 v w^2 +
2 b^6 c^2 u^3 v w^2 - 6 a^4 c^4 u^3 v w^2 -
3 a^2 b^2 c^4 u^3 v w^2 - 5 b^4 c^4 u^3 v w^2 +
4 a^2 c^6 u^3 v w^2 + 4 b^2 c^6 u^3 v w^2 - c^8 u^3 v w^2 -
2 a^8 u^2 v^2 w^2 + 6 a^6 b^2 u^2 v^2 w^2 -
6 a^4 b^4 u^2 v^2 w^2 + 2 a^2 b^6 u^2 v^2 w^2 +
8 a^6 c^2 u^2 v^2 w^2 - 10 a^4 b^2 c^2 u^2 v^2 w^2 +
2 b^6 c^2 u^2 v^2 w^2 - 12 a^4 c^4 u^2 v^2 w^2 -
10 a^2 b^2 c^4 u^2 v^2 w^2 - 6 b^4 c^4 u^2 v^2 w^2 +
8 a^2 c^6 u^2 v^2 w^2 + 6 b^2 c^6 u^2 v^2 w^2 -
2 c^8 u^2 v^2 w^2 - a^8 u v^3 w^2 + 3 a^6 b^2 u v^3 w^2 -
3 a^4 b^4 u v^3 w^2 + a^2 b^6 u v^3 w^2 + 4 a^6 c^2 u v^3 w^2 -
6 a^4 b^2 c^2 u v^3 w^2 + 2 a^2 b^4 c^2 u v^3 w^2 -
7 a^4 c^4 u v^3 w^2 - 7 a^2 b^2 c^4 u v^3 w^2 +
4 a^2 c^6 u v^3 w^2 + a^6 b^2 u^3 w^3 - 2 a^4 b^4 u^3 w^3 +
a^2 b^6 u^3 w^3 - a^4 b^2 c^2 u^3 w^3 + b^6 c^2 u^3 w^3 -
a^2 b^2 c^4 u^3 w^3 - 2 b^4 c^4 u^3 w^3 + b^2 c^6 u^3 w^3 -
a^8 u^2 v w^3 + 4 a^6 b^2 u^2 v w^3 - 5 a^4 b^4 u^2 v w^3 +
2 a^2 b^6 u^2 v w^3 + 4 a^6 c^2 u^2 v w^3 -
3 a^4 b^2 c^2 u^2 v w^3 - 2 a^2 b^4 c^2 u^2 v w^3 +
b^6 c^2 u^2 v w^3 - 6 a^4 c^4 u^2 v w^3 -
4 a^2 b^2 c^4 u^2 v w^3 - 3 b^4 c^4 u^2 v w^3 +
4 a^2 c^6 u^2 v w^3 + 3 b^2 c^6 u^2 v w^3 - c^8 u^2 v w^3 -
2 a^8 u v^2 w^3 + 5 a^6 b^2 u v^2 w^3 - 4 a^4 b^4 u v^2 w^3 +
a^2 b^6 u v^2 w^3 + 6 a^6 c^2 u v^2 w^3 -
6 a^4 b^2 c^2 u v^2 w^3 - 6 a^4 c^4 u v^2 w^3 -
3 a^2 b^2 c^4 u v^2 w^3 + 2 a^2 c^6 u v^2 w^3 - a^8 v^3 w^3 +
2 a^6 b^2 v^3 w^3 - a^4 b^4 v^3 w^3 + a^4 c^4 v^3 w^3),
c^2 (a^6 c^2 u^3 v^3 - a^4 b^2 c^2 u^3 v^3 - a^2 b^4 c^2 u^3 v^3 +
b^6 c^2 u^3 v^3 - 2 a^4 c^4 u^3 v^3 - 2 b^4 c^4 u^3 v^3 +
a^2 c^6 u^3 v^3 + b^2 c^6 u^3 v^3 - a^8 u^3 v^2 w +
4 a^6 b^2 u^3 v^2 w - 6 a^4 b^4 u^3 v^2 w + 4 a^2 b^6 u^3 v^2 w -
b^8 u^3 v^2 w + 3 a^6 c^2 u^3 v^2 w - 4 a^4 b^2 c^2 u^3 v^2 w -
3 a^2 b^4 c^2 u^3 v^2 w + 4 b^6 c^2 u^3 v^2 w -
3 a^4 c^4 u^3 v^2 w - 2 a^2 b^2 c^4 u^3 v^2 w -
5 b^4 c^4 u^3 v^2 w + a^2 c^6 u^3 v^2 w + 2 b^2 c^6 u^3 v^2 w -
a^8 u^2 v^3 w + 4 a^6 b^2 u^2 v^3 w - 6 a^4 b^4 u^2 v^3 w +
4 a^2 b^6 u^2 v^3 w - b^8 u^2 v^3 w + 4 a^6 c^2 u^2 v^3 w -
3 a^4 b^2 c^2 u^2 v^3 w - 4 a^2 b^4 c^2 u^2 v^3 w +
3 b^6 c^2 u^2 v^3 w - 5 a^4 c^4 u^2 v^3 w -
2 a^2 b^2 c^4 u^2 v^3 w - 3 b^4 c^4 u^2 v^3 w +
2 a^2 c^6 u^2 v^3 w + b^2 c^6 u^2 v^3 w + 2 a^6 b^2 u^3 v w^2 -
6 a^4 b^4 u^3 v w^2 + 6 a^2 b^6 u^3 v w^2 - 2 b^8 u^3 v w^2 -
3 a^4 b^2 c^2 u^3 v w^2 - 6 a^2 b^4 c^2 u^3 v w^2 +
5 b^6 c^2 u^3 v w^2 - 4 b^4 c^4 u^3 v w^2 + b^2 c^6 u^3 v w^2 -
2 a^8 u^2 v^2 w^2 + 8 a^6 b^2 u^2 v^2 w^2 -
12 a^4 b^4 u^2 v^2 w^2 + 8 a^2 b^6 u^2 v^2 w^2 -
2 b^8 u^2 v^2 w^2 + 6 a^6 c^2 u^2 v^2 w^2 -
10 a^4 b^2 c^2 u^2 v^2 w^2 - 10 a^2 b^4 c^2 u^2 v^2 w^2 +
6 b^6 c^2 u^2 v^2 w^2 - 6 a^4 c^4 u^2 v^2 w^2 -
6 b^4 c^4 u^2 v^2 w^2 + 2 a^2 c^6 u^2 v^2 w^2 +
2 b^2 c^6 u^2 v^2 w^2 - 2 a^8 u v^3 w^2 + 6 a^6 b^2 u v^3 w^2 -
6 a^4 b^4 u v^3 w^2 + 2 a^2 b^6 u v^3 w^2 + 5 a^6 c^2 u v^3 w^2 -
6 a^4 b^2 c^2 u v^3 w^2 - 3 a^2 b^4 c^2 u v^3 w^2 -
4 a^4 c^4 u v^3 w^2 + a^2 c^6 u v^3 w^2 + a^4 b^4 u^3 w^3 -
b^8 u^3 w^3 + 2 b^6 c^2 u^3 w^3 - b^4 c^4 u^3 w^3 +
4 a^6 b^2 u^2 v w^3 - 7 a^4 b^4 u^2 v w^3 + 4 a^2 b^6 u^2 v w^3 -
b^8 u^2 v w^3 - 7 a^4 b^2 c^2 u^2 v w^3 -
6 a^2 b^4 c^2 u^2 v w^3 + 3 b^6 c^2 u^2 v w^3 +
2 a^2 b^2 c^4 u^2 v w^3 - 3 b^4 c^4 u^2 v w^3 +
b^2 c^6 u^2 v w^3 - a^8 u v^2 w^3 + 4 a^6 b^2 u v^2 w^3 -
7 a^4 b^4 u v^2 w^3 + 4 a^2 b^6 u v^2 w^3 + 3 a^6 c^2 u v^2 w^3 -
6 a^4 b^2 c^2 u v^2 w^3 - 7 a^2 b^4 c^2 u v^2 w^3 -
3 a^4 c^4 u v^2 w^3 + 2 a^2 b^2 c^4 u v^2 w^3 +
a^2 c^6 u v^2 w^3 - a^8 v^3 w^3 + a^4 b^4 v^3 w^3 +
2 a^6 c^2 v^3 w^3 - a^4 c^4 v^3 w^3)}

For A'B'C' = circuncevian triangle of P=(u:v:w) with respect to ABC the intersection of the three radical axes is the point:

{-a^4 b^2 c^4 u^4 v^2 + 4 a^2 b^4 c^4 u^4 v^2 - 3 b^6 c^4 u^4 v^2 +
4 a^2 b^2 c^6 u^4 v^2 + 6 b^4 c^6 u^4 v^2 - 3 b^2 c^8 u^4 v^2 -
a^6 c^4 u^3 v^3 + 4 a^4 b^2 c^4 u^3 v^3 - a^2 b^4 c^4 u^3 v^3 -
2 b^6 c^4 u^3 v^3 + 4 a^4 c^6 u^3 v^3 + 6 a^2 b^2 c^6 u^3 v^3 +
6 b^4 c^6 u^3 v^3 - 5 a^2 c^8 u^3 v^3 - 6 b^2 c^8 u^3 v^3 +
2 c^10 u^3 v^3 + 2 a^4 b^2 c^4 u^2 v^4 - 2 a^2 b^4 c^4 u^2 v^4 +
2 a^4 c^6 u^2 v^4 + 4 a^2 b^2 c^6 u^2 v^4 - 2 a^2 c^8 u^2 v^4 +
2 a^6 b^2 c^2 u^4 v w - 7 a^4 b^4 c^2 u^4 v w +
8 a^2 b^6 c^2 u^4 v w - 3 b^8 c^2 u^4 v w - 7 a^4 b^2 c^4 u^4 v w +
3 b^6 c^4 u^4 v w + 8 a^2 b^2 c^6 u^4 v w + 3 b^4 c^6 u^4 v w -
3 b^2 c^8 u^4 v w + 2 a^8 c^2 u^3 v^2 w - 7 a^6 b^2 c^2 u^3 v^2 w +
8 a^4 b^4 c^2 u^3 v^2 w - 3 a^2 b^6 c^2 u^3 v^2 w -
8 a^6 c^4 u^3 v^2 w + 4 a^4 b^2 c^4 u^3 v^2 w +
2 a^2 b^4 c^4 u^3 v^2 w - 2 b^6 c^4 u^3 v^2 w +
12 a^4 c^6 u^3 v^2 w + 9 a^2 b^2 c^6 u^3 v^2 w +
6 b^4 c^6 u^3 v^2 w - 8 a^2 c^8 u^3 v^2 w - 6 b^2 c^8 u^3 v^2 w +
2 c^10 u^3 v^2 w + a^4 b^4 c^2 u^2 v^3 w - 2 a^2 b^6 c^2 u^2 v^3 w +
b^8 c^2 u^2 v^3 w + 6 a^4 b^2 c^4 u^2 v^3 w +
2 a^2 b^4 c^4 u^2 v^3 w - 4 b^6 c^4 u^2 v^3 w + a^4 c^6 u^2 v^3 w +
2 a^2 b^2 c^6 u^2 v^3 w + 6 b^4 c^6 u^2 v^3 w -
2 a^2 c^8 u^2 v^3 w - 4 b^2 c^8 u^2 v^3 w + c^10 u^2 v^3 w +
a^6 b^2 c^2 u v^4 w - 2 a^4 b^4 c^2 u v^4 w + a^2 b^6 c^2 u v^4 w +
3 a^6 c^4 u v^4 w + 4 a^4 b^2 c^4 u v^4 w - 3 a^2 b^4 c^4 u v^4 w -
2 a^4 c^6 u v^4 w + 3 a^2 b^2 c^6 u v^4 w - a^2 c^8 u v^4 w -
a^4 b^4 c^2 u^4 w^2 + 4 a^2 b^6 c^2 u^4 w^2 - 3 b^8 c^2 u^4 w^2 +
4 a^2 b^4 c^4 u^4 w^2 + 6 b^6 c^4 u^4 w^2 - 3 b^4 c^6 u^4 w^2 +
2 a^8 b^2 u^3 v w^2 - 8 a^6 b^4 u^3 v w^2 + 12 a^4 b^6 u^3 v w^2 -
8 a^2 b^8 u^3 v w^2 + 2 b^10 u^3 v w^2 - 7 a^6 b^2 c^2 u^3 v w^2 +
4 a^4 b^4 c^2 u^3 v w^2 + 9 a^2 b^6 c^2 u^3 v w^2 -
6 b^8 c^2 u^3 v w^2 + 8 a^4 b^2 c^4 u^3 v w^2 +
2 a^2 b^4 c^4 u^3 v w^2 + 6 b^6 c^4 u^3 v w^2 -
3 a^2 b^2 c^6 u^3 v w^2 - 2 b^4 c^6 u^3 v w^2 +
2 a^10 u^2 v^2 w^2 - 7 a^8 b^2 u^2 v^2 w^2 +
8 a^6 b^4 u^2 v^2 w^2 - 2 a^4 b^6 u^2 v^2 w^2 -
2 a^2 b^8 u^2 v^2 w^2 + b^10 u^2 v^2 w^2 - 7 a^8 c^2 u^2 v^2 w^2 +
6 a^4 b^4 c^2 u^2 v^2 w^2 + 4 a^2 b^6 c^2 u^2 v^2 w^2 -
3 b^8 c^2 u^2 v^2 w^2 + 8 a^6 c^4 u^2 v^2 w^2 +
6 a^4 b^2 c^4 u^2 v^2 w^2 - 4 a^2 b^4 c^4 u^2 v^2 w^2 +
2 b^6 c^4 u^2 v^2 w^2 - 2 a^4 c^6 u^2 v^2 w^2 +
4 a^2 b^2 c^6 u^2 v^2 w^2 + 2 b^4 c^6 u^2 v^2 w^2 -
2 a^2 c^8 u^2 v^2 w^2 - 3 b^2 c^8 u^2 v^2 w^2 + c^10 u^2 v^2 w^2 +
a^10 u v^3 w^2 - 4 a^8 b^2 u v^3 w^2 + 6 a^6 b^4 u v^3 w^2 -
4 a^4 b^6 u v^3 w^2 + a^2 b^8 u v^3 w^2 - 4 a^8 c^2 u v^3 w^2 +
a^6 b^2 c^2 u v^3 w^2 + 6 a^4 b^4 c^2 u v^3 w^2 -
3 a^2 b^6 c^2 u v^3 w^2 + 5 a^6 c^4 u v^3 w^2 +
3 a^2 b^4 c^4 u v^3 w^2 - 2 a^4 c^6 u v^3 w^2 -
a^2 b^2 c^6 u v^3 w^2 + a^8 c^2 v^4 w^2 - a^4 b^4 c^2 v^4 w^2 +
2 a^4 b^2 c^4 v^4 w^2 - a^4 c^6 v^4 w^2 - a^6 b^4 u^3 w^3 +
4 a^4 b^6 u^3 w^3 - 5 a^2 b^8 u^3 w^3 + 2 b^10 u^3 w^3 +
4 a^4 b^4 c^2 u^3 w^3 + 6 a^2 b^6 c^2 u^3 w^3 - 6 b^8 c^2 u^3 w^3 -
a^2 b^4 c^4 u^3 w^3 + 6 b^6 c^4 u^3 w^3 - 2 b^4 c^6 u^3 w^3 +
a^4 b^6 u^2 v w^3 - 2 a^2 b^8 u^2 v w^3 + b^10 u^2 v w^3 +
6 a^4 b^4 c^2 u^2 v w^3 + 2 a^2 b^6 c^2 u^2 v w^3 -
4 b^8 c^2 u^2 v w^3 + a^4 b^2 c^4 u^2 v w^3 +
2 a^2 b^4 c^4 u^2 v w^3 + 6 b^6 c^4 u^2 v w^3 -
2 a^2 b^2 c^6 u^2 v w^3 - 4 b^4 c^6 u^2 v w^3 + b^2 c^8 u^2 v w^3 +
a^10 u v^2 w^3 - 4 a^8 b^2 u v^2 w^3 + 5 a^6 b^4 u v^2 w^3 -
2 a^4 b^6 u v^2 w^3 - 4 a^8 c^2 u v^2 w^3 + a^6 b^2 c^2 u v^2 w^3 -
a^2 b^6 c^2 u v^2 w^3 + 6 a^6 c^4 u v^2 w^3 +
6 a^4 b^2 c^4 u v^2 w^3 + 3 a^2 b^4 c^4 u v^2 w^3 -
4 a^4 c^6 u v^2 w^3 - 3 a^2 b^2 c^6 u v^2 w^3 + a^2 c^8 u v^2 w^3 -
a^8 b^2 v^3 w^3 + 2 a^6 b^4 v^3 w^3 - a^4 b^6 v^3 w^3 -
a^8 c^2 v^3 w^3 - 4 a^6 b^2 c^2 v^3 w^3 + a^4 b^4 c^2 v^3 w^3 +
2 a^6 c^4 v^3 w^3 + a^4 b^2 c^4 v^3 w^3 - a^4 c^6 v^3 w^3 +
2 a^4 b^6 u^2 w^4 - 2 a^2 b^8 u^2 w^4 + 2 a^4 b^4 c^2 u^2 w^4 +
4 a^2 b^6 c^2 u^2 w^4 - 2 a^2 b^4 c^4 u^2 w^4 + 3 a^6 b^4 u v w^4 -
2 a^4 b^6 u v w^4 - a^2 b^8 u v w^4 + a^6 b^2 c^2 u v w^4 +
4 a^4 b^4 c^2 u v w^4 + 3 a^2 b^6 c^2 u v w^4 -
2 a^4 b^2 c^4 u v w^4 - 3 a^2 b^4 c^4 u v w^4 +
a^2 b^2 c^6 u v w^4 + a^8 b^2 v^2 w^4 - a^4 b^6 v^2 w^4 +
2 a^4 b^4 c^2 v^2 w^4 -
a^4 b^2 c^4 v^2 w^4, -2 a^4 b^2 c^4 u^4 v^2 +
2 a^2 b^4 c^4 u^4 v^2 + 4 a^2 b^2 c^6 u^4 v^2 + 2 b^4 c^6 u^4 v^2 -
2 b^2 c^8 u^4 v^2 - 2 a^6 c^4 u^3 v^3 - a^4 b^2 c^4 u^3 v^3 +
4 a^2 b^4 c^4 u^3 v^3 - b^6 c^4 u^3 v^3 + 6 a^4 c^6 u^3 v^3 +
6 a^2 b^2 c^6 u^3 v^3 + 4 b^4 c^6 u^3 v^3 - 6 a^2 c^8 u^3 v^3 -
5 b^2 c^8 u^3 v^3 + 2 c^10 u^3 v^3 - 3 a^6 c^4 u^2 v^4 +
4 a^4 b^2 c^4 u^2 v^4 - a^2 b^4 c^4 u^2 v^4 + 6 a^4 c^6 u^2 v^4 +
4 a^2 b^2 c^6 u^2 v^4 - 3 a^2 c^8 u^2 v^4 + a^6 b^2 c^2 u^4 v w -
2 a^4 b^4 c^2 u^4 v w + a^2 b^6 c^2 u^4 v w -
3 a^4 b^2 c^4 u^4 v w + 4 a^2 b^4 c^4 u^4 v w + 3 b^6 c^4 u^4 v w +
3 a^2 b^2 c^6 u^4 v w - 2 b^4 c^6 u^4 v w - b^2 c^8 u^4 v w +
a^8 c^2 u^3 v^2 w - 2 a^6 b^2 c^2 u^3 v^2 w +
a^4 b^4 c^2 u^3 v^2 w - 4 a^6 c^4 u^3 v^2 w +
2 a^4 b^2 c^4 u^3 v^2 w + 6 a^2 b^4 c^4 u^3 v^2 w +
6 a^4 c^6 u^3 v^2 w + 2 a^2 b^2 c^6 u^3 v^2 w + b^4 c^6 u^3 v^2 w -
4 a^2 c^8 u^3 v^2 w - 2 b^2 c^8 u^3 v^2 w + c^10 u^3 v^2 w -
3 a^6 b^2 c^2 u^2 v^3 w + 8 a^4 b^4 c^2 u^2 v^3 w -
7 a^2 b^6 c^2 u^2 v^3 w + 2 b^8 c^2 u^2 v^3 w -
2 a^6 c^4 u^2 v^3 w + 2 a^4 b^2 c^4 u^2 v^3 w +
4 a^2 b^4 c^4 u^2 v^3 w - 8 b^6 c^4 u^2 v^3 w +
6 a^4 c^6 u^2 v^3 w + 9 a^2 b^2 c^6 u^2 v^3 w +
12 b^4 c^6 u^2 v^3 w - 6 a^2 c^8 u^2 v^3 w - 8 b^2 c^8 u^2 v^3 w +
2 c^10 u^2 v^3 w - 3 a^8 c^2 u v^4 w + 8 a^6 b^2 c^2 u v^4 w -
7 a^4 b^4 c^2 u v^4 w + 2 a^2 b^6 c^2 u v^4 w + 3 a^6 c^4 u v^4 w -
7 a^2 b^4 c^4 u v^4 w + 3 a^4 c^6 u v^4 w + 8 a^2 b^2 c^6 u v^4 w -
3 a^2 c^8 u v^4 w - a^4 b^4 c^2 u^4 w^2 + b^8 c^2 u^4 w^2 +
2 a^2 b^4 c^4 u^4 w^2 - b^4 c^6 u^4 w^2 + a^8 b^2 u^3 v w^2 -
4 a^6 b^4 u^3 v w^2 + 6 a^4 b^6 u^3 v w^2 - 4 a^2 b^8 u^3 v w^2 +
b^10 u^3 v w^2 - 3 a^6 b^2 c^2 u^3 v w^2 + 6 a^4 b^4 c^2 u^3 v w^2 +
a^2 b^6 c^2 u^3 v w^2 - 4 b^8 c^2 u^3 v w^2 +
3 a^4 b^2 c^4 u^3 v w^2 + 5 b^6 c^4 u^3 v w^2 -
a^2 b^2 c^6 u^3 v w^2 - 2 b^4 c^6 u^3 v w^2 + a^10 u^2 v^2 w^2 -
2 a^8 b^2 u^2 v^2 w^2 - 2 a^6 b^4 u^2 v^2 w^2 +
8 a^4 b^6 u^2 v^2 w^2 - 7 a^2 b^8 u^2 v^2 w^2 +
2 b^10 u^2 v^2 w^2 - 3 a^8 c^2 u^2 v^2 w^2 +
4 a^6 b^2 c^2 u^2 v^2 w^2 + 6 a^4 b^4 c^2 u^2 v^2 w^2 -
7 b^8 c^2 u^2 v^2 w^2 + 2 a^6 c^4 u^2 v^2 w^2 -
4 a^4 b^2 c^4 u^2 v^2 w^2 + 6 a^2 b^4 c^4 u^2 v^2 w^2 +
8 b^6 c^4 u^2 v^2 w^2 + 2 a^4 c^6 u^2 v^2 w^2 +
4 a^2 b^2 c^6 u^2 v^2 w^2 - 2 b^4 c^6 u^2 v^2 w^2 -
3 a^2 c^8 u^2 v^2 w^2 - 2 b^2 c^8 u^2 v^2 w^2 + c^10 u^2 v^2 w^2 +
2 a^10 u v^3 w^2 - 8 a^8 b^2 u v^3 w^2 + 12 a^6 b^4 u v^3 w^2 -
8 a^4 b^6 u v^3 w^2 + 2 a^2 b^8 u v^3 w^2 - 6 a^8 c^2 u v^3 w^2 +
9 a^6 b^2 c^2 u v^3 w^2 + 4 a^4 b^4 c^2 u v^3 w^2 -
7 a^2 b^6 c^2 u v^3 w^2 + 6 a^6 c^4 u v^3 w^2 +
2 a^4 b^2 c^4 u v^3 w^2 + 8 a^2 b^4 c^4 u v^3 w^2 -
2 a^4 c^6 u v^3 w^2 - 3 a^2 b^2 c^6 u v^3 w^2 - 3 a^8 c^2 v^4 w^2 +
4 a^6 b^2 c^2 v^4 w^2 - a^4 b^4 c^2 v^4 w^2 + 6 a^6 c^4 v^4 w^2 +
4 a^4 b^2 c^4 v^4 w^2 - 3 a^4 c^6 v^4 w^2 - a^6 b^4 u^3 w^3 +
2 a^4 b^6 u^3 w^3 - a^2 b^8 u^3 w^3 + a^4 b^4 c^2 u^3 w^3 -
4 a^2 b^6 c^2 u^3 w^3 - b^8 c^2 u^3 w^3 + a^2 b^4 c^4 u^3 w^3 +
2 b^6 c^4 u^3 w^3 - b^4 c^6 u^3 w^3 - 2 a^6 b^4 u^2 v w^3 +
5 a^4 b^6 u^2 v w^3 - 4 a^2 b^8 u^2 v w^3 + b^10 u^2 v w^3 -
a^6 b^2 c^2 u^2 v w^3 + a^2 b^6 c^2 u^2 v w^3 -
4 b^8 c^2 u^2 v w^3 + 3 a^4 b^2 c^4 u^2 v w^3 +
6 a^2 b^4 c^4 u^2 v w^3 + 6 b^6 c^4 u^2 v w^3 -
3 a^2 b^2 c^6 u^2 v w^3 - 4 b^4 c^6 u^2 v w^3 + b^2 c^8 u^2 v w^3 +
a^10 u v^2 w^3 - 2 a^8 b^2 u v^2 w^3 + a^6 b^4 u v^2 w^3 -
4 a^8 c^2 u v^2 w^3 + 2 a^6 b^2 c^2 u v^2 w^3 +
6 a^4 b^4 c^2 u v^2 w^3 + 6 a^6 c^4 u v^2 w^3 +
2 a^4 b^2 c^4 u v^2 w^3 + a^2 b^4 c^4 u v^2 w^3 -
4 a^4 c^6 u v^2 w^3 - 2 a^2 b^2 c^6 u v^2 w^3 + a^2 c^8 u v^2 w^3 +
2 a^10 v^3 w^3 - 5 a^8 b^2 v^3 w^3 + 4 a^6 b^4 v^3 w^3 -
a^4 b^6 v^3 w^3 - 6 a^8 c^2 v^3 w^3 + 6 a^6 b^2 c^2 v^3 w^3 +
4 a^4 b^4 c^2 v^3 w^3 + 6 a^6 c^4 v^3 w^3 - a^4 b^2 c^4 v^3 w^3 -
2 a^4 c^6 v^3 w^3 - a^6 b^4 u^2 w^4 + a^2 b^8 u^2 w^4 +
2 a^4 b^4 c^2 u^2 w^4 - a^2 b^4 c^4 u^2 w^4 - a^8 b^2 u v w^4 -
2 a^6 b^4 u v w^4 + 3 a^4 b^6 u v w^4 + 3 a^6 b^2 c^2 u v w^4 +
4 a^4 b^4 c^2 u v w^4 + a^2 b^6 c^2 u v w^4 -
3 a^4 b^2 c^4 u v w^4 - 2 a^2 b^4 c^4 u v w^4 +
a^2 b^2 c^6 u v w^4 - 2 a^8 b^2 v^2 w^4 + 2 a^6 b^4 v^2 w^4 +
4 a^6 b^2 c^2 v^2 w^4 + 2 a^4 b^4 c^2 v^2 w^4 -
2 a^4 b^2 c^4 v^2 w^4, -a^4 b^2 c^4 u^4 v^2 +
2 a^2 b^4 c^4 u^4 v^2 - b^6 c^4 u^4 v^2 + b^2 c^8 u^4 v^2 -
a^6 c^4 u^3 v^3 + a^4 b^2 c^4 u^3 v^3 + a^2 b^4 c^4 u^3 v^3 -
b^6 c^4 u^3 v^3 + 2 a^4 c^6 u^3 v^3 - 4 a^2 b^2 c^6 u^3 v^3 +
2 b^4 c^6 u^3 v^3 - a^2 c^8 u^3 v^3 - b^2 c^8 u^3 v^3 -
a^6 c^4 u^2 v^4 + 2 a^4 b^2 c^4 u^2 v^4 - a^2 b^4 c^4 u^2 v^4 +
a^2 c^8 u^2 v^4 + a^6 b^2 c^2 u^4 v w - 3 a^4 b^4 c^2 u^4 v w +
3 a^2 b^6 c^2 u^4 v w - b^8 c^2 u^4 v w - 2 a^4 b^2 c^4 u^4 v w +
4 a^2 b^4 c^4 u^4 v w - 2 b^6 c^4 u^4 v w + a^2 b^2 c^6 u^4 v w +
3 b^4 c^6 u^4 v w + a^8 c^2 u^3 v^2 w - 3 a^6 b^2 c^2 u^3 v^2 w +
3 a^4 b^4 c^2 u^3 v^2 w - a^2 b^6 c^2 u^3 v^2 w -
4 a^6 c^4 u^3 v^2 w + 6 a^4 b^2 c^4 u^3 v^2 w -
2 b^6 c^4 u^3 v^2 w + 6 a^4 c^6 u^3 v^2 w + a^2 b^2 c^6 u^3 v^2 w +
5 b^4 c^6 u^3 v^2 w - 4 a^2 c^8 u^3 v^2 w - 4 b^2 c^8 u^3 v^2 w +
c^10 u^3 v^2 w - a^6 b^2 c^2 u^2 v^3 w + 3 a^4 b^4 c^2 u^2 v^3 w -
3 a^2 b^6 c^2 u^2 v^3 w + b^8 c^2 u^2 v^3 w - 2 a^6 c^4 u^2 v^3 w +
6 a^2 b^4 c^4 u^2 v^3 w - 4 b^6 c^4 u^2 v^3 w +
5 a^4 c^6 u^2 v^3 w + a^2 b^2 c^6 u^2 v^3 w + 6 b^4 c^6 u^2 v^3 w -
4 a^2 c^8 u^2 v^3 w - 4 b^2 c^8 u^2 v^3 w + c^10 u^2 v^3 w -
a^8 c^2 u v^4 w + 3 a^6 b^2 c^2 u v^4 w - 3 a^4 b^4 c^2 u v^4 w +
a^2 b^6 c^2 u v^4 w - 2 a^6 c^4 u v^4 w + 4 a^4 b^2 c^4 u v^4 w -
2 a^2 b^4 c^4 u v^4 w + 3 a^4 c^6 u v^4 w + a^2 b^2 c^6 u v^4 w -
2 a^4 b^4 c^2 u^4 w^2 + 4 a^2 b^6 c^2 u^4 w^2 - 2 b^8 c^2 u^4 w^2 +
2 a^2 b^4 c^4 u^4 w^2 + 2 b^6 c^4 u^4 w^2 + a^8 b^2 u^3 v w^2 -
4 a^6 b^4 u^3 v w^2 + 6 a^4 b^6 u^3 v w^2 - 4 a^2 b^8 u^3 v w^2 +
b^10 u^3 v w^2 - 2 a^6 b^2 c^2 u^3 v w^2 +
2 a^4 b^4 c^2 u^3 v w^2 + 2 a^2 b^6 c^2 u^3 v w^2 -
2 b^8 c^2 u^3 v w^2 + a^4 b^2 c^4 u^3 v w^2 +
6 a^2 b^4 c^4 u^3 v w^2 + b^6 c^4 u^3 v w^2 + a^10 u^2 v^2 w^2 -
3 a^8 b^2 u^2 v^2 w^2 + 2 a^6 b^4 u^2 v^2 w^2 +
2 a^4 b^6 u^2 v^2 w^2 - 3 a^2 b^8 u^2 v^2 w^2 + b^10 u^2 v^2 w^2 -
2 a^8 c^2 u^2 v^2 w^2 + 4 a^6 b^2 c^2 u^2 v^2 w^2 -
4 a^4 b^4 c^2 u^2 v^2 w^2 + 4 a^2 b^6 c^2 u^2 v^2 w^2 -
2 b^8 c^2 u^2 v^2 w^2 - 2 a^6 c^4 u^2 v^2 w^2 +
6 a^4 b^2 c^4 u^2 v^2 w^2 + 6 a^2 b^4 c^4 u^2 v^2 w^2 -
2 b^6 c^4 u^2 v^2 w^2 + 8 a^4 c^6 u^2 v^2 w^2 +
8 b^4 c^6 u^2 v^2 w^2 - 7 a^2 c^8 u^2 v^2 w^2 -
7 b^2 c^8 u^2 v^2 w^2 + 2 c^10 u^2 v^2 w^2 + a^10 u v^3 w^2 -
4 a^8 b^2 u v^3 w^2 + 6 a^6 b^4 u v^3 w^2 - 4 a^4 b^6 u v^3 w^2 +
a^2 b^8 u v^3 w^2 - 2 a^8 c^2 u v^3 w^2 + 2 a^6 b^2 c^2 u v^3 w^2 +
2 a^4 b^4 c^2 u v^3 w^2 - 2 a^2 b^6 c^2 u v^3 w^2 +
a^6 c^4 u v^3 w^2 + 6 a^4 b^2 c^4 u v^3 w^2 +
a^2 b^4 c^4 u v^3 w^2 - 2 a^8 c^2 v^4 w^2 + 4 a^6 b^2 c^2 v^4 w^2 -
2 a^4 b^4 c^2 v^4 w^2 + 2 a^6 c^4 v^4 w^2 + 2 a^4 b^2 c^4 v^4 w^2 -
2 a^6 b^4 u^3 w^3 + 6 a^4 b^6 u^3 w^3 - 6 a^2 b^8 u^3 w^3 +
2 b^10 u^3 w^3 - a^4 b^4 c^2 u^3 w^3 + 6 a^2 b^6 c^2 u^3 w^3 -
5 b^8 c^2 u^3 w^3 + 4 a^2 b^4 c^4 u^3 w^3 + 4 b^6 c^4 u^3 w^3 -
b^4 c^6 u^3 w^3 - 2 a^6 b^4 u^2 v w^3 + 6 a^4 b^6 u^2 v w^3 -
6 a^2 b^8 u^2 v w^3 + 2 b^10 u^2 v w^3 - 3 a^6 b^2 c^2 u^2 v w^3 +
2 a^4 b^4 c^2 u^2 v w^3 + 9 a^2 b^6 c^2 u^2 v w^3 -
8 b^8 c^2 u^2 v w^3 + 8 a^4 b^2 c^4 u^2 v w^3 +
4 a^2 b^4 c^4 u^2 v w^3 + 12 b^6 c^4 u^2 v w^3 -
7 a^2 b^2 c^6 u^2 v w^3 - 8 b^4 c^6 u^2 v w^3 +
2 b^2 c^8 u^2 v w^3 + 2 a^10 u v^2 w^3 - 6 a^8 b^2 u v^2 w^3 +
6 a^6 b^4 u v^2 w^3 - 2 a^4 b^6 u v^2 w^3 - 8 a^8 c^2 u v^2 w^3 +
9 a^6 b^2 c^2 u v^2 w^3 + 2 a^4 b^4 c^2 u v^2 w^3 -
3 a^2 b^6 c^2 u v^2 w^3 + 12 a^6 c^4 u v^2 w^3 +
4 a^4 b^2 c^4 u v^2 w^3 + 8 a^2 b^4 c^4 u v^2 w^3 -
8 a^4 c^6 u v^2 w^3 - 7 a^2 b^2 c^6 u v^2 w^3 +
2 a^2 c^8 u v^2 w^3 + 2 a^10 v^3 w^3 - 6 a^8 b^2 v^3 w^3 +
6 a^6 b^4 v^3 w^3 - 2 a^4 b^6 v^3 w^3 - 5 a^8 c^2 v^3 w^3 +
6 a^6 b^2 c^2 v^3 w^3 - a^4 b^4 c^2 v^3 w^3 + 4 a^6 c^4 v^3 w^3 +
4 a^4 b^2 c^4 v^3 w^3 - a^4 c^6 v^3 w^3 - 3 a^6 b^4 u^2 w^4 +
6 a^4 b^6 u^2 w^4 - 3 a^2 b^8 u^2 w^4 + 4 a^4 b^4 c^2 u^2 w^4 +
4 a^2 b^6 c^2 u^2 w^4 - a^2 b^4 c^4 u^2 w^4 - 3 a^8 b^2 u v w^4 +
3 a^6 b^4 u v w^4 + 3 a^4 b^6 u v w^4 - 3 a^2 b^8 u v w^4 +
8 a^6 b^2 c^2 u v w^4 + 8 a^2 b^6 c^2 u v w^4 -
7 a^4 b^2 c^4 u v w^4 - 7 a^2 b^4 c^4 u v w^4 +
2 a^2 b^2 c^6 u v w^4 - 3 a^8 b^2 v^2 w^4 + 6 a^6 b^4 v^2 w^4 -
3 a^4 b^6 v^2 w^4 + 4 a^6 b^2 c^2 v^2 w^4 + 4 a^4 b^4 c^2 v^2 w^4 -
a^4 b^2 c^4 v^2 w^4}

Francisco Javier García Capitán
15 December 2011

------

Generalization:

Let 123456 be a cyclic hexagon (ie inscribed in a circle).

The radical axes of the NPCs of the pairs of the triangles:

abc, def, where {a,b,c,d,e,f} = {1,2,3,4,5,6} (ie pairs of triangles with no common vertex) are concurrent (at the common midpoint of the distances of the centers of the pairs of the NPCs).

We have 10 pairs of triangles:

(123,456), (124,356), (125,346), (126,345)

(134,256), (135,246), (136,245)

(145,236), (146,235)

(156,234)

The centers of the 10 NPCs lie on a conic centered at the point of concurrence of the radical axes.

APH, 17 December 2011

NINE POINT CIRCLE

Let ABC be a triangle, A'B'C' the orthic triangle and P a point.

Let A*,B*,C* be the orthogonal projections of A,B,C on the line OP, resp.
Let L1,L2,L3 be the reflections of A'A*,B'B*,C'C* in the altitudes AA',BB',CC', resp. and M1,M2,M3 the parallels through A,B,C, to L1,L2,L3, resp.

The lines M1,M2,M3 concur at a point Q on the Nine Point Circle of ABC (Q is the center of the rectangular circumhyperbola which is the isogonal conjugate of the line OP)

APH, 9 December 2011