Τρίτη, 15 Απριλίου 2014

CIRCUMCENTERS ON THE LINES OH (Euler Line), OI

Let ABC be a triangle and

1. A'B'C' the pedal triangle of H (orthic triangle)

Denote:

(Oa) = the circumcircle of OBC.

(O1) = the reflection of (Oa) in BC.

(O'1) = the reflection of (O1) in HA'.

(Ob) = the circumcircle of OCA.

(O2) = the reflection of (Ob) in CA.

(O'2) = the reflection of (O2) in HB'.

(Oc) = the circumcircle of OAB.

(O3) = the reflection of (Oc) in AB.

(O'3) = the reflection of (O3) in HC'.

The circumcenter of the triangle O'1O'2O'3 lies on the OH line (Euler line)

-----

2. A'B'C' the pedal triangle of I.

Denote:

(Oa) = the circumcircle of IBC.

(O1) = the reflection of (Oa) in BC.

(O'1) = the reflection of (O1) in IA'.

(Ob) = the circumcircle of ICA.

(O2) = the reflection of (Ob) in CA.

(O'2) = the reflection of (O2) in IB'.

(Oc) = the circumcircle of IAB.

(O3) = the reflection of (Oc) in AB.

(O'3) = the reflection of (O3) in IC'.

The circumcenter of the triangle O'1O'2O'3 lies on the OI line

Generalizations (Loci):

Let ABC be a triangle, P,P* two isogonal conjugate points and A'B'C',A"B"C" the pedal triangles of P,P*.

Denote:

(Oa) = the circumcircle of PBC.

(O1) = the reflection of (Oa) in BC.

(O'1) = the reflection of (O1) in PA'.

(O"1) = the reflection of (O1) in P*A"

(Ob) = the circumcircle of PCA.

(O2) = the reflection of (Ob) in CA.

(O'2) = the reflection of (O2) in PB'.

(O"2) = the reflection of (O2) in P*B"

(Oc) = the circumcircle of PAB.

(O3) = the reflection of (Oc) in AB.

(O'3) = the reflection of (O3) in PC'.

(O"3) = the reflection of (O3) in P*C"

R' = the circumcenter of O'O'2O'3

R" = the circumcenter of O"1O"2O"3

Which is the locus of P such that:

1. O, P, R'

2. O, P, R"

3. O, R', R"

4. P, P*, R'

5. P, R', R"

are collinear ?

The McCay cubic?

Antreas P. Hatzipolakis, 16 April 2014.

RADICAL CENTERS ON THE LINES OH (Euler line),OI

Let ABC be a triangle and

1. A'B'C' the pedal triangle of H (orthic triangle)

Denote:

(Oa) = the circumcircle of OBC.

(O1) = the reflection of (Oa) in BC.

(O'1) = the reflection of (O1) in HA'.

(Ob) = the circumcircle of OCA.

(O2) = the reflection of (Ob) in CA.

(O'2) = the reflection of (O2) in HB'.

(Oc) = the circumcircle of OAB.

(O3) = the reflection of (Oc) in AB.

(O'3) = the reflection of (O3) in HC'.

The radical center of (O'1),(O'2),(O'3) lies on the OH line (Euler line)

-----

2. A'B'C' the pedal triangle of I.

Denote:

(Oa) = the circumcircle of IBC.

(O1) = the reflection of (Oa) in BC.

(O'1) = the reflection of (O1) in IA'.

(Ob) = the circumcircle of ICA.

(O2) = the reflection of (Ob) in CA.

(O'2) = the reflection of (O2) in IB'.

(Oc) = the circumcircle of IAB.

(O3) = the reflection of (Oc) in AB.

(O'3) = the reflection of (O3) in IC'.

The radical center of (O'1),(O'2),(O'3) lies on the OI line.

Generalizations (Loci):

Let ABC be a triangle, P,P* two isogonal conjugate points and A'B'C',A"B"C" the pedal triangles of P,P*.

Denote:

(Oa) = the circumcircle of PBC.

(O1) = the reflection of (Oa) in BC.

(O'1) = the reflection of (O1) in PA'.

(O"1) = the reflection of (O1) in P*A"

(Ob) = the circumcircle of PCA.

(O2) = the reflection of (Ob) in CA.

(O'2) = the reflection of (O2) in PB'.

(O"2) = the reflection of (O2) in P*B"

(Oc) = the circumcircle of PAB.

(O3) = the reflection of (Oc) in AB.

(O'3) = the reflection of (O3) in PC'.

(O"3) = the reflection of (O3) in P*C"

R' = the radical center of (O'1),(O'2),(O'3)

R" = the radical center of (O"1),(O"2),(O"3)

Which is the locus of P such that:

1. O, P, R'

2. O, P, R"

3. O, R', R"

4. P, P*, R'

5. P, R', R"

are collinear ?

The McCay cubic?

Antreas P. Hatzipolakis, 15 April 2014.

Παρασκευή, 11 Απριλίου 2014

CONCURRENT CIRCLES -- EULER LINE

Theorem:

Let ABC be a triangle and A',B',C' three points. If the circumcircles of A'BC, B'CA, C'AB are concurrent, then also the circumcircles of AB'C', BC'A', CA'B' are concurrent.

Corollary:

Let ABC be a triangle and P a point. If A',B',C' are arbitrary points on the circumcircles of PBC,PCA,PAB, resp. then the circumcircles of AB'C', BC'A', CA'B' are concurrent.

Applications:

Let P,Q be two points and PaPbPc, Q1Q2Q3 the antipedal, pedal triangles of P,Q, resp. The orthogonal projections A',B',C' of Pa,Pb,Pc on PQ1,PQ2,PQ3, resp. lie on the circumcircles of PBC,PCA,PAB, resp.

The circumcircles of AB'C', BC'A', CA'B' concur at a point D.

1. For P = O, Q = H:

The point D lies on the Euler line of ABC

2. For P = Q = I:

The point D lies on the Euler line of ABC

3. For P = Q = H:

The Euler lines L,L1,L2,L3 of ABC, DBC,DCA,DAB are concurrent at a point D' on the Neuberg cubic.

The parallels to L1,L2,L3 through A,B,C, resp. are concurrent at a point D"

Coordinates of the points D's, D' and D"?

More pairs (P,Q) such that the circumcircles concur on the Euler line?

Antreas P. Hatzipolakis, 11 April 2014

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

It seems that the general case for D has truly appalling barycentrics.

But, for the cases you mention we have:

1) PQ = OH, D= X(186)

2) PQ = II, D = X(1325)

3) PQ = HH, D = X(1157)

Also PQ = OI, D = X(36)

Peter Moses, 14 April 2014


Πέμπτη, 10 Απριλίου 2014

RADICAL AXES 1.1

Let ABC be a triangle, P a point and A'B'C' the cevian triangle of P. Denote:

Ab, Ac = the reflections of A' in AB, AC. resp.

A2, A3 = the reflections of A' in BB', CC', resp.

Oab, Oac = the circumcenters of BAbA2, CAcA3, resp.

Similarly (cyclically):

Obc, Oba and Oca, Ocb.

1. P = O.

The radical axes R1 =:((Oab),(Oac)), R2 =:((Obc),(Oba)), R3 =:((Oca),(Ocb))are concurrent at O.

The reflections of R1,R2,R3 in BC,CA,AB are the lines AN,BN,CN, resp.

2. P = H.

The radical axes R1 =:((Oab),(Oac)), R2 =:((Obc),(Oba)), R3 =:((Oca),(Ocb)) are concurrent on the Euler line of ABC.

The reflections of the radical axes S1 =:((Obc), (Ocb)), S2 =:((Oca),(Oac)), S3 =: ((Oab), (Oba)) in BC,CA,AB, resp. are concurrent.

The reflections of the radical axes T1 =: ((Oba), (Oca)), T2 =:((Ocb),(Oab)), T3 =:((Oac), (Obc)) in Bc,CA,AB are concurrent.

The triangles: bounded by the lines (T1,T2,T3) and the orthic A'B'C' are paralle;ogic.

Antreas P. Hatzipolakis, 10 April 2014


Παρασκευή, 6 Δεκεμβρίου 2013

CONCURRENT EULER LINES

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

Denote:

A1, Ab, Ac = the midpoints of A'P, A'B, A'C, resp.

B2, Bc, Ba = the midpoints of B'P, B'C, B'A, resp.

C3, Ca, Cb = the midpoints of C'P, C'A, C'B, resp.

For P = O the Euler lines of the triangles A1AbAc, B2BcBa, C3CaCb are concurrent (trivial case)

How about for P = I ??

In general, which is the locus of P such that the Euler lines of the triangles A1AbAc, B2BcBa, C3CaCb are concurrent?

Antreas P. Hatzipolakis, 5 Dec. 2013, Anopolis #1143

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

The locus is a degree-7 circum-excentral-curve which passes through ETC-centers I, O and X(1138)=Isogonal conjugate of X(399)

1) For P=X(3)=O, point of concurrence is Z=O

2) For P=X(1138), Z=X(30)

3) For P=X(1)=I

Z = (2*a^4-2*(b+c)*a^3-(3*b^2+4*b*c+3*c^2)*a^2+2*(c^2-3*b*c+b^2)*(b+c)*a+(b^2-c^2)^2)/a : : (Trilinears)

= midpoint of: (1,442), (21,3649)

= on lines (1,442), (7,21), (30,551), (78,3826), (79,5426), (191,3338), (497,2475), (758,942), (950,3838), (958,3487), (962,4428), (1387,3636), (2646,5249), (3035,3812), (3651,5603), (3671,4640), (3897,5434)

= [2.022889134016317502091, 1.585022278218526006039, 1.609700227440945328741]

César Lozada, 6 Dec. 2013, Anopolis #1144

Παρασκευή, 29 Νοεμβρίου 2013

PERSPECTIVITY - LOCUS

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

Denote:

Ab = the other than B intersection of the circle with diameter BA' and the circumcircle.

Ac = the other than C intersection of the circle with diameter CA' and the circumcircle.

Similarly (cyclically): Bc, Ba and Ca, Cb

Which is the locus of P such that:

1. ABC, Triangle bounded by (AbAc, BcBa, CaCb) are perspective?

Answer: The entire plane. Reference.

2. ABC, Triangle bounded by (BcCb, CaAc, AbBa) are perspective? The entire plane?

Special case: BcCb, CaAc, AbBa are concurrent.

3. Triangle bounded by (AbAc, BcBa, CaCb) and Triangle bounded by (BcCb, CaAc, AbBa) are perspective?

Antreas P. Hatzipolakis, 29 Nov. 2013

______________________________________

Which is the locus of P such that:

1. ABC, Triangle bounded by (AbAc, BcBa, CaCb) are perspective?

Answer: The entire plane.

**** Perspector, if P=(u:v:w) (baricentric coordinates):

Q =( a^2 (a^2 - b^2 - c^2)/(-2 a^2 c^2 u^2 v^2 + 2 b^2 c^2 u^2 v^2 + 2 c^4 u^2 v^2 + a^4 u^2 v w - 2 a^2 b^2 u^2 v w + b^4 u^2 v w - 2 a^2 c^2 u^2 v w + 6 b^2 c^2 u^2 v w + c^4 u^2 v w - 2 a^2 b^2 u^2 w^2 + 2 b^4 u^2 w^2 + 2 b^2 c^2 u^2 w^2 + 2 a^4 v^2 w^2 - 2 a^2 b^2 v^2 w^2 - 2 a^2 c^2 v^2 w^2) : ... : ...)

Pairs of triangle centres {P, Q}: {X(69),X(64)}, {X(99),X(520)}

(Reference)

2. ABC, Triangle bounded by (BcCb, CaAc, AbBa) are perspective?

**** If P lies on Darboux cubic.

Special case: BcCb, CaAc, AbBa are concurrent.

**** If P lies on the sextic (No centers ETC):

-2 a^8 c^4 x^3 y^3 + 4 a^4 b^4 c^4 x^3 y^3 - 2 b^8 c^4 x^3 y^3 + 4 a^6 c^6 x^3 y^3 - 4 a^4 b^2 c^6 x^3 y^3 - 4 a^2 b^4 c^6 x^3 y^3 + 4 b^6 c^6 x^3 y^3 + 8 a^2 b^2 c^8 x^3 y^3 - 4 a^2 c^10 x^3 y^3 - 4 b^2 c^10 x^3 y^3 + 2 c^12 x^3 y^3 + a^10 c^2 x^3 y^2 z - 3 a^8 b^2 c^2 x^3 y^2 z + 2 a^6 b^4 c^2 x^3 y^2 z + 2 a^4 b^6 c^2 x^3 y^2 z - 3 a^2 b^8 c^2 x^3 y^2 z + b^10 c^2 x^3 y^2 z - 3 a^8 c^4 x^3 y^2 z + 8 a^6 b^2 c^4 x^3 y^2 z + 2 a^4 b^4 c^4 x^3 y^2 z - 7 b^8 c^4 x^3 y^2 z + 2 a^6 c^6 x^3 y^2 z - 6 a^4 b^2 c^6 x^3 y^2 z + 6 a^2 b^4 c^6 x^3 y^2 z + 14 b^6 c^6 x^3 y^2 z + 2 a^4 c^8 x^3 y^2 z - 10 b^4 c^8 x^3 y^2 z - 3 a^2 c^10 x^3 y^2 z + b^2 c^10 x^3 y^2 z + c^12 x^3 y^2 z + a^10 c^2 x^2 y^3 z - 3 a^8 b^2 c^2 x^2 y^3 z + 2 a^6 b^4 c^2 x^2 y^3 z + 2 a^4 b^6 c^2 x^2 y^3 z - 3 a^2 b^8 c^2 x^2 y^3 z + b^10 c^2 x^2 y^3 z - 7 a^8 c^4 x^2 y^3 z + 2 a^4 b^4 c^4 x^2 y^3 z + 8 a^2 b^6 c^4 x^2 y^3 z - 3 b^8 c^4 x^2 y^3 z + 14 a^6 c^6 x^2 y^3 z + 6 a^4 b^2 c^6 x^2 y^3 z - 6 a^2 b^4 c^6 x^2 y^3 z + 2 b^6 c^6 x^2 y^3 z - 10 a^4 c^8 x^2 y^3 z + 2 b^4 c^8 x^2 y^3 z + a^2 c^10 x^2 y^3 z - 3 b^2 c^10 x^2 y^3 z + c^12 x^2 y^3 z + a^10 b^2 x^3 y z^2 - 3 a^8 b^4 x^3 y z^2 + 2 a^6 b^6 x^3 y z^2 + 2 a^4 b^8 x^3 y z^2 - 3 a^2 b^10 x^3 y z^2 + b^12 x^3 y z^2 - 3 a^8 b^2 c^2 x^3 y z^2 + 8 a^6 b^4 c^2 x^3 y z^2 - 6 a^4 b^6 c^2 x^3 y z^2 + b^10 c^2 x^3 y z^2 + 2 a^6 b^2 c^4 x^3 y z^2 + 2 a^4 b^4 c^4 x^3 y z^2 + 6 a^2 b^6 c^4 x^3 y z^2 - 10 b^8 c^4 x^3 y z^2 + 2 a^4 b^2 c^6 x^3 y z^2 + 14 b^6 c^6 x^3 y z^2 - 3 a^2 b^2 c^8 x^3 y z^2 - 7 b^4 c^8 x^3 y z^2 + b^2 c^10 x^3 y z^2 + a^12 x^2 y^2 z^2 - 2 a^10 b^2 x^2 y^2 z^2 - a^8 b^4 x^2 y^2 z^2 + 4 a^6 b^6 x^2 y^2 z^2 - a^4 b^8 x^2 y^2 z^2 - 2 a^2 b^10 x^2 y^2 z^2 + b^12 x^2 y^2 z^2 - 2 a^10 c^2 x^2 y^2 z^2 + 6 a^8 b^2 c^2 x^2 y^2 z^2 - 4 a^6 b^4 c^2 x^2 y^2 z^2 - 4 a^4 b^6 c^2 x^2 y^2 z^2 + 6 a^2 b^8 c^2 x^2 y^2 z^2 - 2 b^10 c^2 x^2 y^2 z^2 - a^8 c^4 x^2 y^2 z^2 - 4 a^6 b^2 c^4 x^2 y^2 z^2 + 10 a^4 b^4 c^4 x^2 y^2 z^2 - 4 a^2 b^6 c^4 x^2 y^2 z^2 - b^8 c^4 x^2 y^2 z^2 + 4 a^6 c^6 x^2 y^2 z^2 - 4 a^4 b^2 c^6 x^2 y^2 z^2 - 4 a^2 b^4 c^6 x^2 y^2 z^2 + 4 b^6 c^6 x^2 y^2 z^2 - a^4 c^8 x^2 y^2 z^2 + 6 a^2 b^2 c^8 x^2 y^2 z^2 - b^4 c^8 x^2 y^2 z^2 - 2 a^2 c^10 x^2 y^2 z^2 - 2 b^2 c^10 x^2 y^2 z^2 + c^12 x^2 y^2 z^2 + a^12 x y^3 z^2 - 3 a^10 b^2 x y^3 z^2 + 2 a^8 b^4 x y^3 z^2 + 2 a^6 b^6 x y^3 z^2 - 3 a^4 b^8 x y^3 z^2 + a^2 b^10 x y^3 z^2 + a^10 c^2 x y^3 z^2 - 6 a^6 b^4 c^2 x y^3 z^2 + 8 a^4 b^6 c^2 x y^3 z^2 - 3 a^2 b^8 c^2 x y^3 z^2 - 10 a^8 c^4 x y^3 z^2 + 6 a^6 b^2 c^4 x y^3 z^2 + 2 a^4 b^4 c^4 x y^3 z^2 + 2 a^2 b^6 c^4 x y^3 z^2 + 14 a^6 c^6 x y^3 z^2 + 2 a^2 b^4 c^6 x y^3 z^2 - 7 a^4 c^8 x y^3 z^2 - 3 a^2 b^2 c^8 x y^3 z^2 + a^2 c^10 x y^3 z^2 - 2 a^8 b^4 x^3 z^3 + 4 a^6 b^6 x^3 z^3 - 4 a^2 b^10 x^3 z^3 + 2 b^12 x^3 z^3 - 4 a^4 b^6 c^2 x^3 z^3 + 8 a^2 b^8 c^2 x^3 z^3 - 4 b^10 c^2 x^3 z^3 + 4 a^4 b^4 c^4 x^3 z^3 - 4 a^2 b^6 c^4 x^3 z^3 + 4 b^6 c^6 x^3 z^3 - 2 b^4 c^8 x^3 z^3 + a^10 b^2 x^2 y z^3 - 7 a^8 b^4 x^2 y z^3 + 14 a^6 b^6 x^2 y z^3 - 10 a^4 b^8 x^2 y z^3 + a^2 b^10 x^2 y z^3 + b^12 x^2 y z^3 - 3 a^8 b^2 c^2 x^2 y z^3 + 6 a^4 b^6 c^2 x^2 y z^3 - 3 b^10 c^2 x^2 y z^3 + 2 a^6 b^2 c^4 x^2 y z^3 + 2 a^4 b^4 c^4 x^2 y z^3 - 6 a^2 b^6 c^4 x^2 y z^3 + 2 b^8 c^4 x^2 y z^3 + 2 a^4 b^2 c^6 x^2 y z^3 + 8 a^2 b^4 c^6 x^2 y z^3 + 2 b^6 c^6 x^2 y z^3 - 3 a^2 b^2 c^8 x^2 y z^3 - 3 b^4 c^8 x^2 y z^3 + b^2 c^10 x^2 y z^3 + a^12 x y^2 z^3 + a^10 b^2 x y^2 z^3 - 10 a^8 b^4 x y^2 z^3 + 14 a^6 b^6 x y^2 z^3 - 7 a^4 b^8 x y^2 z^3 + a^2 b^10 x y^2 z^3 - 3 a^10 c^2 x y^2 z^3 + 6 a^6 b^4 c^2 x y^2 z^3 - 3 a^2 b^8 c^2 x y^2 z^3 + 2 a^8 c^4 x y^2 z^3 - 6 a^6 b^2 c^4 x y^2 z^3 + 2 a^4 b^4 c^4 x y^2 z^3 + 2 a^2 b^6 c^4 x y^2 z^3 + 2 a^6 c^6 x y^2 z^3 + 8 a^4 b^2 c^6 x y^2 z^3 + 2 a^2 b^4 c^6 x y^2 z^3 - 3 a^4 c^8 x y^2 z^3 - 3 a^2 b^2 c^8 x y^2 z^3 + a^2 c^10 x y^2 z^3 + 2 a^12 y^3 z^3 - 4 a^10 b^2 y^3 z^3 + 4 a^6 b^6 y^3 z^3 - 2 a^4 b^8 y^3 z^3 - 4 a^10 c^2 y^3 z^3 + 8 a^8 b^2 c^2 y^3 z^3 - 4 a^6 b^4 c^2 y^3 z^3 - 4 a^6 b^2 c^4 y^3 z^3 + 4 a^4 b^4 c^4 y^3 z^3 + 4 a^6 c^6 y^3 z^3 - 2 a^4 c^8 y^3 z^3=0

3. Triangle bounded by (AbAc, BcBa, CaCb) and Triangle bounded by (BcCb, CaAc, AbBa) are perspective?

**** Always. The coordinates of perspector is very complicated.

First barycentric coordinate:

a^2 (32 a^14 b^2 c^8 u^7 v^5+96 a^12 b^4 c^8 u^7 v^5-480 a^10 b^6 c^8 u^7 v^5+352 a^8 b^8 c^8 u^7 v^5+352 a^6 b^10 c^8 u^7 v^5-480 a^4 b^12 c^8 u^7 v^5+96 a^2 b^14 c^8 u^7 v^5+32 b^16 c^8 u^7 v^5-96 a^12 b^2 c^10 u^7 v^5+64 a^10 b^4 c^10 u^7 v^5+608 a^8 b^6 c^10 u^7 v^5-1152 a^6 b^8 c^10 u^7 v^5+608 a^4 b^10 c^10 u^7 v^5+64 a^2 b^12 c^10 u^7 v^5-96 b^14 c^10 u^7 v^5+32 a^10 b^2 c^12 u^7 v^5-608 a^8 b^4 c^12 u^7 v^5+576 a^6 b^6 c^12 u^7 v^5+576 a^4 b^8 c^12 u^7 v^5-608 a^2 b^10 c^12 u^7 v^5+32 b^12 c^12 u^7 v^5+160 a^8 b^2 c^14 u^7 v^5+384 a^6 b^4 c^14 u^7 v^5-1088 a^4 b^6 c^14 u^7 v^5+384 a^2 b^8 c^14 u^7 v^5+160 b^10 c^14 u^7 v^5-160 a^6 b^2 c^16 u^7 v^5+416 a^4 b^4 c^16 u^7 v^5+416 a^2 b^6 c^16 u^7 v^5-160 b^8 c^16 u^7 v^5-32 a^4 b^2 c^18 u^7 v^5-448 a^2 b^4 c^18 u^7 v^5-32 b^6 c^18 u^7 v^5+96 a^2 b^2 c^20 u^7 v^5+96 b^4 c^20 u^7 v^5-32 b^2 c^22 u^7 v^5+64 a^14 b^2 c^8 u^6 v^6-128 a^12 b^4 c^8 u^6 v^6-64 a^10 b^6 c^8 u^6 v^6+256 a^8 b^8 c^8 u^6 v^6-64 a^6 b^10 c^8 u^6 v^6-128 a^4 b^12 c^8 u^6 v^6+64 a^2 b^14 c^8 u^6 v^6-128 a^12 b^2 c^10 u^6 v^6+384 a^10 b^4 c^10 u^6 v^6-256 a^8 b^6 c^10 u^6 v^6-256 a^6 b^8 c^10 u^6 v^6+384 a^4 b^10 c^10 u^6 v^6-128 a^2 b^12 c^10 u^6 v^6-64 a^10 b^2 c^12 u^6 v^6-256 a^8 b^4 c^12 u^6 v^6+640 a^6 b^6 c^12 u^6 v^6-256 a^4 b^8 c^12 u^6 v^6-64 a^2 b^10 c^12 u^6 v^6+256 a^8 b^2 c^14 u^6 v^6-256 a^6 b^4 c^14 u^6 v^6-256 a^4 b^6 c^14 u^6 v^6+256 a^2 b^8 c^14 u^6 v^6-64 a^6 b^2 c^16 u^6 v^6+384 a^4 b^4 c^16 u^6 v^6-64 a^2 b^6 c^16 u^6 v^6-128 a^4 b^2 c^18 u^6 v^6-128 a^2 b^4 c^18 u^6 v^6+64 a^2 b^2 c^20 u^6 v^6-32 a^16 b^2 c^6 u^7 v^4 w-32 a^14 b^4 c^6 u^7 v^4 w+480 a^12 b^6 c^6 u^7 v^4 w-800 a^10 b^8 c^6 u^7 v^4 w+160 a^8 b^10 c^6 u^7 v^4 w+672 a^6 b^12 c^6 u^7 v^4 w-608 a^4 b^14 c^6 u^7 v^4 w+160 a^2 b^16 c^6 u^7 v^4 w+128 a^14 b^2 c^8 u^7 v^4 w-96 a^12 b^4 c^8 u^7 v^4 w-1472 a^10 b^6 c^8 u^7 v^4 w+2912 a^8 b^8 c^8 u^7 v^4 w-1024 a^6 b^10 c^8 u^7 v^4 w-928 a^4 b^12 c^8 u^7 v^4 w+320 a^2 b^14 c^8 u^7 v^4 w+160 b^16 c^8 u^7 v^4 w-128 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a^14 b^10 u^3 v^4 w^5+252 a^12 b^12 u^3 v^4 w^5-212 a^10 b^14 u^3 v^4 w^5+74 a^8 b^16 u^3 v^4 w^5-14 a^6 b^18 u^3 v^4 w^5+6 a^4 b^20 u^3 v^4 w^5-2 a^2 b^22 u^3 v^4 w^5-10 a^22 c^2 u^3 v^4 w^5-76 a^20 b^2 c^2 u^3 v^4 w^5+318 a^18 b^4 c^2 u^3 v^4 w^5-240 a^16 b^6 c^2 u^3 v^4 w^5-340 a^14 b^8 c^2 u^3 v^4 w^5+728 a^12 b^10 c^2 u^3 v^4 w^5-724 a^10 b^12 c^2 u^3 v^4 w^5+592 a^8 b^14 c^2 u^3 v^4 w^5-290 a^6 b^16 c^2 u^3 v^4 w^5+20 a^4 b^18 c^2 u^3 v^4 w^5+22 a^2 b^20 c^2 u^3 v^4 w^5+14 a^20 c^4 u^3 v^4 w^5+194 a^18 b^2 c^4 u^3 v^4 w^5-616 a^16 b^4 c^4 u^3 v^4 w^5-1560 a^14 b^6 c^4 u^3 v^4 w^5+5124 a^12 b^8 c^4 u^3 v^4 w^5-2788 a^10 b^10 c^4 u^3 v^4 w^5-1928 a^8 b^12 c^4 u^3 v^4 w^5+1672 a^6 b^14 c^4 u^3 v^4 w^5-34 a^4 b^16 c^4 u^3 v^4 w^5-78 a^2 b^18 c^4 u^3 v^4 w^5+10 a^18 c^6 u^3 v^4 w^5-176 a^16 b^2 c^6 u^3 v^4 w^5+664 a^14 b^4 c^6 u^3 v^4 w^5+336 a^12 b^6 c^6 u^3 v^4 w^5-868 a^10 b^8 c^6 u^3 v^4 w^5+2928 a^8 b^10 c^6 u^3 v^4 w^5-2728 a^6 b^12 c^6 u^3 v^4 w^5-272 a^4 b^14 c^6 u^3 v^4 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u^3 v^4 w^5+2 a^2 c^22 u^3 v^4 w^5+4 a^24 u^2 v^5 w^5-16 a^22 b^2 u^2 v^5 w^5+12 a^20 b^4 u^2 v^5 w^5+32 a^18 b^6 u^2 v^5 w^5-56 a^16 b^8 u^2 v^5 w^5+56 a^12 b^12 u^2 v^5 w^5-32 a^10 b^14 u^2 v^5 w^5-12 a^8 b^16 u^2 v^5 w^5+16 a^6 b^18 u^2 v^5 w^5-4 a^4 b^20 u^2 v^5 w^5-16 a^22 c^2 u^2 v^5 w^5+8 a^20 b^2 c^2 u^2 v^5 w^5-32 a^18 b^4 c^2 u^2 v^5 w^5+352 a^16 b^6 c^2 u^2 v^5 w^5-512 a^14 b^8 c^2 u^2 v^5 w^5-144 a^12 b^10 c^2 u^2 v^5 w^5+672 a^10 b^12 c^2 u^2 v^5 w^5-288 a^8 b^14 c^2 u^2 v^5 w^5-112 a^6 b^16 c^2 u^2 v^5 w^5+72 a^4 b^18 c^2 u^2 v^5 w^5+12 a^20 c^4 u^2 v^5 w^5-32 a^18 b^2 c^4 u^2 v^5 w^5-1296 a^16 b^4 c^4 u^2 v^5 w^5+2304 a^14 b^6 c^4 u^2 v^5 w^5+1160 a^12 b^8 c^4 u^2 v^5 w^5-3872 a^10 b^10 c^4 u^2 v^5 w^5+1392 a^8 b^12 c^4 u^2 v^5 w^5+576 a^6 b^14 c^4 u^2 v^5 w^5-244 a^4 b^16 c^4 u^2 v^5 w^5+32 a^18 c^6 u^2 v^5 w^5+352 a^16 b^2 c^6 u^2 v^5 w^5+2304 a^14 b^4 c^6 u^2 v^5 w^5-5216 a^12 b^6 c^6 u^2 v^5 w^5+3232 a^10 b^8 c^6 u^2 v^5 w^5+288 a^8 b^10 c^6 u^2 v^5 w^5-1216 a^6 b^12 c^6 u^2 v^5 w^5+224 a^4 b^14 c^6 u^2 v^5 w^5-56 a^16 c^8 u^2 v^5 w^5-512 a^14 b^2 c^8 u^2 v^5 w^5+1160 a^12 b^4 c^8 u^2 v^5 w^5+3232 a^10 b^6 c^8 u^2 v^5 w^5-2760 a^8 b^8 c^8 u^2 v^5 w^5+736 a^6 b^10 c^8 u^2 v^5 w^5+248 a^4 b^12 c^8 u^2 v^5 w^5-144 a^12 b^2 c^10 u^2 v^5 w^5-3872 a^10 b^4 c^10 u^2 v^5 w^5+288 a^8 b^6 c^10 u^2 v^5 w^5+736 a^6 b^8 c^10 u^2 v^5 w^5-592 a^4 b^10 c^10 u^2 v^5 w^5+56 a^12 c^12 u^2 v^5 w^5+672 a^10 b^2 c^12 u^2 v^5 w^5+1392 a^8 b^4 c^12 u^2 v^5 w^5-1216 a^6 b^6 c^12 u^2 v^5 w^5+248 a^4 b^8 c^12 u^2 v^5 w^5-32 a^10 c^14 u^2 v^5 w^5-288 a^8 b^2 c^14 u^2 v^5 w^5+576 a^6 b^4 c^14 u^2 v^5 w^5+224 a^4 b^6 c^14 u^2 v^5 w^5-12 a^8 c^16 u^2 v^5 w^5-112 a^6 b^2 c^16 u^2 v^5 w^5-244 a^4 b^4 c^16 u^2 v^5 w^5+16 a^6 c^18 u^2 v^5 w^5+72 a^4 b^2 c^18 u^2 v^5 w^5-4 a^4 c^20 u^2 v^5 w^5-64 a^20 b^2 c^2 u v^6 w^5+192 a^18 b^4 c^2 u v^6 w^5-64 a^16 b^6 c^2 u v^6 w^5-320 a^14 b^8 c^2 u v^6 w^5+320 a^12 b^10 c^2 u v^6 w^5+64 a^10 b^12 c^2 u v^6 w^5-192 a^8 b^14 c^2 u v^6 w^5+64 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c^4 u^6 w^6-256 a^8 b^12 c^4 u^6 w^6-256 a^6 b^14 c^4 u^6 w^6+384 a^4 b^16 c^4 u^6 w^6-128 a^2 b^18 c^4 u^6 w^6-64 a^10 b^8 c^6 u^6 w^6-256 a^8 b^10 c^6 u^6 w^6+640 a^6 b^12 c^6 u^6 w^6-256 a^4 b^14 c^6 u^6 w^6-64 a^2 b^16 c^6 u^6 w^6+256 a^8 b^8 c^8 u^6 w^6-256 a^6 b^10 c^8 u^6 w^6-256 a^4 b^12 c^8 u^6 w^6+256 a^2 b^14 c^8 u^6 w^6-64 a^6 b^8 c^10 u^6 w^6+384 a^4 b^10 c^10 u^6 w^6-64 a^2 b^12 c^10 u^6 w^6-128 a^4 b^8 c^12 u^6 w^6-128 a^2 b^10 c^12 u^6 w^6+64 a^2 b^8 c^14 u^6 w^6+64 a^16 b^6 c^2 u^5 v w^6+192 a^14 b^8 c^2 u^5 v w^6-960 a^12 b^10 c^2 u^5 v w^6+704 a^10 b^12 c^2 u^5 v w^6+704 a^8 b^14 c^2 u^5 v w^6-960 a^6 b^16 c^2 u^5 v w^6+192 a^4 b^18 c^2 u^5 v w^6+64 a^2 b^20 c^2 u^5 v w^6-192 a^14 b^6 c^4 u^5 v w^6+128 a^12 b^8 c^4 u^5 v w^6+1216 a^10 b^10 c^4 u^5 v w^6-2304 a^8 b^12 c^4 u^5 v w^6+1216 a^6 b^14 c^4 u^5 v w^6+128 a^4 b^16 c^4 u^5 v w^6-192 a^2 b^18 c^4 u^5 v w^6+64 a^12 b^6 c^6 u^5 v w^6-1216 a^10 b^8 c^6 u^5 v w^6+1152 a^8 b^10 c^6 u^5 v w^6+1152 a^6 b^12 c^6 u^5 v w^6-1216 a^4 b^14 c^6 u^5 v w^6+64 a^2 b^16 c^6 u^5 v w^6+320 a^10 b^6 c^8 u^5 v w^6+768 a^8 b^8 c^8 u^5 v w^6-2176 a^6 b^10 c^8 u^5 v w^6+768 a^4 b^12 c^8 u^5 v w^6+320 a^2 b^14 c^8 u^5 v w^6-320 a^8 b^6 c^10 u^5 v w^6+832 a^6 b^8 c^10 u^5 v w^6+832 a^4 b^10 c^10 u^5 v w^6-320 a^2 b^12 c^10 u^5 v w^6-64 a^6 b^6 c^12 u^5 v w^6-896 a^4 b^8 c^12 u^5 v w^6-64 a^2 b^10 c^12 u^5 v w^6+192 a^4 b^6 c^14 u^5 v w^6+192 a^2 b^8 c^14 u^5 v w^6-64 a^2 b^6 c^16 u^5 v w^6+16 a^18 b^4 c^2 u^4 v^2 w^6+192 a^16 b^6 c^2 u^4 v^2 w^6-192 a^14 b^8 c^2 u^4 v^2 w^6-1216 a^12 b^10 c^2 u^4 v^2 w^6+2400 a^10 b^12 c^2 u^4 v^2 w^6-1216 a^8 b^14 c^2 u^4 v^2 w^6-192 a^6 b^16 c^2 u^4 v^2 w^6+192 a^4 b^18 c^2 u^4 v^2 w^6+16 a^2 b^20 c^2 u^4 v^2 w^6-64 a^16 b^4 c^4 u^4 v^2 w^6-320 a^14 b^6 c^4 u^4 v^2 w^6+1344 a^12 b^8 c^4 u^4 v^2 w^6-960 a^10 b^10 c^4 u^4 v^2 w^6-960 a^8 b^12 c^4 u^4 v^2 w^6+1344 a^6 b^14 c^4 u^4 v^2 w^6-320 a^4 b^16 c^4 u^4 v^2 w^6-64 a^2 b^18 c^4 u^4 v^2 w^6+64 a^14 b^4 c^6 u^4 v^2 w^6-448 a^12 b^6 c^6 u^4 v^2 w^6-1344 a^10 b^8 c^6 u^4 v^2 w^6+3456 a^8 b^10 c^6 u^4 v^2 w^6-1344 a^6 b^12 c^6 u^4 v^2 w^6-448 a^4 b^14 c^6 u^4 v^2 w^6+64 a^2 b^16 c^6 u^4 v^2 w^6+64 a^12 b^4 c^8 u^4 v^2 w^6+1088 a^10 b^6 c^8 u^4 v^2 w^6-1152 a^8 b^8 c^8 u^4 v^2 w^6-1152 a^6 b^10 c^8 u^4 v^2 w^6+1088 a^4 b^12 c^8 u^4 v^2 w^6+64 a^2 b^14 c^8 u^4 v^2 w^6-160 a^10 b^4 c^10 u^4 v^2 w^6-192 a^8 b^6 c^10 u^4 v^2 w^6+1984 a^6 b^8 c^10 u^4 v^2 w^6-192 a^4 b^10 c^10 u^4 v^2 w^6-160 a^2 b^12 c^10 u^4 v^2 w^6+64 a^8 b^4 c^12 u^4 v^2 w^6-704 a^6 b^6 c^12 u^4 v^2 w^6-704 a^4 b^8 c^12 u^4 v^2 w^6+64 a^2 b^10 c^12 u^4 v^2 w^6+64 a^6 b^4 c^14 u^4 v^2 w^6+448 a^4 b^6 c^14 u^4 v^2 w^6+64 a^2 b^8 c^14 u^4 v^2 w^6-64 a^4 b^4 c^16 u^4 v^2 w^6-64 a^2 b^6 c^16 u^4 v^2 w^6+16 a^2 b^4 c^18 u^4 v^2 w^6-16 a^20 b^2 c^2 u^2 v^4 w^6-192 a^18 b^4 c^2 u^2 v^4 w^6+192 a^16 b^6 c^2 u^2 v^4 w^6+1216 a^14 b^8 c^2 u^2 v^4 w^6-2400 a^12 b^10 c^2 u^2 v^4 w^6+1216 a^10 b^12 c^2 u^2 v^4 w^6+192 a^8 b^14 c^2 u^2 v^4 w^6-192 a^6 b^16 c^2 u^2 v^4 w^6-16 a^4 b^18 c^2 u^2 v^4 w^6+64 a^18 b^2 c^4 u^2 v^4 w^6+320 a^16 b^4 c^4 u^2 v^4 w^6-1344 a^14 b^6 c^4 u^2 v^4 w^6+960 a^12 b^8 c^4 u^2 v^4 w^6+960 a^10 b^10 c^4 u^2 v^4 w^6-1344 a^8 b^12 c^4 u^2 v^4 w^6+320 a^6 b^14 c^4 u^2 v^4 w^6+64 a^4 b^16 c^4 u^2 v^4 w^6-64 a^16 b^2 c^6 u^2 v^4 w^6+448 a^14 b^4 c^6 u^2 v^4 w^6+1344 a^12 b^6 c^6 u^2 v^4 w^6-3456 a^10 b^8 c^6 u^2 v^4 w^6+1344 a^8 b^10 c^6 u^2 v^4 w^6+448 a^6 b^12 c^6 u^2 v^4 w^6-64 a^4 b^14 c^6 u^2 v^4 w^6-64 a^14 b^2 c^8 u^2 v^4 w^6-1088 a^12 b^4 c^8 u^2 v^4 w^6+1152 a^10 b^6 c^8 u^2 v^4 w^6+1152 a^8 b^8 c^8 u^2 v^4 w^6-1088 a^6 b^10 c^8 u^2 v^4 w^6-64 a^4 b^12 c^8 u^2 v^4 w^6+160 a^12 b^2 c^10 u^2 v^4 w^6+192 a^10 b^4 c^10 u^2 v^4 w^6-1984 a^8 b^6 c^10 u^2 v^4 w^6+192 a^6 b^8 c^10 u^2 v^4 w^6+160 a^4 b^10 c^10 u^2 v^4 w^6-64 a^10 b^2 c^12 u^2 v^4 w^6+704 a^8 b^4 c^12 u^2 v^4 w^6+704 a^6 b^6 c^12 u^2 v^4 w^6-64 a^4 b^8 c^12 u^2 v^4 w^6-64 a^8 b^2 c^14 u^2 v^4 w^6-448 a^6 b^4 c^14 u^2 v^4 w^6-64 a^4 b^6 c^14 u^2 v^4 w^6+64 a^6 b^2 c^16 u^2 v^4 w^6+64 a^4 b^4 c^16 u^2 v^4 w^6-16 a^4 b^2 c^18 u^2 v^4 w^6-64 a^20 b^2 c^2 u v^5 w^6-192 a^18 b^4 c^2 u v^5 w^6+960 a^16 b^6 c^2 u v^5 w^6-704 a^14 b^8 c^2 u v^5 w^6-704 a^12 b^10 c^2 u v^5 w^6+960 a^10 b^12 c^2 u v^5 w^6-192 a^8 b^14 c^2 u v^5 w^6-64 a^6 b^16 c^2 u v^5 w^6+192 a^18 b^2 c^4 u v^5 w^6-128 a^16 b^4 c^4 u v^5 w^6-1216 a^14 b^6 c^4 u v^5 w^6+2304 a^12 b^8 c^4 u v^5 w^6-1216 a^10 b^10 c^4 u v^5 w^6-128 a^8 b^12 c^4 u v^5 w^6+192 a^6 b^14 c^4 u v^5 w^6-64 a^16 b^2 c^6 u v^5 w^6+1216 a^14 b^4 c^6 u v^5 w^6-1152 a^12 b^6 c^6 u v^5 w^6-1152 a^10 b^8 c^6 u v^5 w^6+1216 a^8 b^10 c^6 u v^5 w^6-64 a^6 b^12 c^6 u v^5 w^6-320 a^14 b^2 c^8 u v^5 w^6-768 a^12 b^4 c^8 u v^5 w^6+2176 a^10 b^6 c^8 u v^5 w^6-768 a^8 b^8 c^8 u v^5 w^6-320 a^6 b^10 c^8 u v^5 w^6+320 a^12 b^2 c^10 u v^5 w^6-832 a^10 b^4 c^10 u v^5 w^6-832 a^8 b^6 c^10 u v^5 w^6+320 a^6 b^8 c^10 u v^5 w^6+64 a^10 b^2 c^12 u v^5 w^6+896 a^8 b^4 c^12 u v^5 w^6+64 a^6 b^6 c^12 u v^5 w^6-192 a^8 b^2 c^14 u v^5 w^6-192 a^6 b^4 c^14 u v^5 w^6+64 a^6 b^2 c^16 u v^5 w^6-64 a^20 b^2 c^2 v^6 w^6+128 a^18 b^4 c^2 v^6 w^6+64 a^16 b^6 c^2 v^6 w^6-256 a^14 b^8 c^2 v^6 w^6+64 a^12 b^10 c^2 v^6 w^6+128 a^10 b^12 c^2 v^6 w^6-64 a^8 b^14 c^2 v^6 w^6+128 a^18 b^2 c^4 v^6 w^6-384 a^16 b^4 c^4 v^6 w^6+256 a^14 b^6 c^4 v^6 w^6+256 a^12 b^8 c^4 v^6 w^6-384 a^10 b^10 c^4 v^6 w^6+128 a^8 b^12 c^4 v^6 w^6+64 a^16 b^2 c^6 v^6 w^6+256 a^14 b^4 c^6 v^6 w^6-640 a^12 b^6 c^6 v^6 w^6+256 a^10 b^8 c^6 v^6 w^6+64 a^8 b^10 c^6 v^6 w^6-256 a^14 b^2 c^8 v^6 w^6+256 a^12 b^4 c^8 v^6 w^6+256 a^10 b^6 c^8 v^6 w^6-256 a^8 b^8 c^8 v^6 w^6+64 a^12 b^2 c^10 v^6 w^6-384 a^10 b^4 c^10 v^6 w^6+64 a^8 b^6 c^10 v^6 w^6+128 a^10 b^2 c^12 v^6 w^6+128 a^8 b^4 c^12 v^6 w^6-64 a^8 b^2 c^14 v^6 w^6)

Angel Montesdeoca, 30 Nov. 2013

ADDENDUM:

4. Triangle ABC, Triangle bounded by (BaCa, CbAb, AcBc) are perspective?

5. Triangle bounded by (BcCb, CaAc, AbBa),Triangle bounded by (BaCa, CbAb, AcBc) are perspective?

6. Triangle bounded by (AbAc, BcBa, CaCb),Triangle bounded by (BaCa, CbAb, AcBc) are perspective?

Antreas P. Hatzipolakis, 30 Nov. 2013


Σάββατο, 14 Σεπτεμβρίου 2013

PRIZE (Re: ORTHOCENTER - REFLECTIONS - CONCURRENT CIRCLES)

[APH]: In fact we can take any point P (instead of H) and any points O1,O2,O3 on the circumcircles of PBC,PCA,PAB, resp.

Then the circumcircles of the triangles

AO2O3, BO3O1, CO1O2

are concurrent.

Anopolis #850

For a proof I offer the book:

R. G. SANGER: SYNTHETIC PROJECTIVE GEOMETRY (1939)

APH

 

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