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
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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)}
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 a^12 b^2 c^10 u^7 v^4 w+480 a^10 b^4 c^10 u^7 v^4 w-480 a^8 b^6 c^10 u^7 v^4 w-2112 a^6 b^8 c^10 u^7 v^4 w+3776 a^4 b^10 c^10 u^7 v^4 w-928 a^2 b^12 c^10 u^7 v^4 w-608 b^14 c^10 u^7 v^4 w-128 a^10 b^2 c^12 u^7 v^4 w-352 a^8 b^4 c^12 u^7 v^4 w+2944 a^6 b^6 c^12 u^7 v^4 w-2112 a^4 b^8 c^12 u^7 v^4 w-1024 a^2 b^10 c^12 u^7 v^4 w+672 b^12 c^12 u^7 v^4 w+320 a^8 b^2 c^14 u^7 v^4 w-352 a^6 b^4 c^14 u^7 v^4 w-480 a^4 b^6 c^14 u^7 v^4 w+2912 a^2 b^8 c^14 u^7 v^4 w+160 b^10 c^14 u^7 v^4 w-128 a^6 b^2 c^16 u^7 v^4 w+480 a^4 b^4 c^16 u^7 v^4 w-1472 a^2 b^6 c^16 u^7 v^4 w-800 b^8 c^16 u^7 v^4 w-128 a^4 b^2 c^18 u^7 v^4 w-96 a^2 b^4 c^18 u^7 v^4 w+480 b^6 c^18 u^7 v^4 w+128 a^2 b^2 c^20 u^7 v^4 w-32 b^4 c^20 u^7 v^4 w-32 b^2 c^22 u^7 v^4 w-8 a^18 c^6 u^6 v^5 w-56 a^16 b^2 c^6 u^6 v^5 w+384 a^14 b^4 c^6 u^6 v^5 w-704 a^12 b^6 c^6 u^6 v^5 w+304 a^10 b^8 c^6 u^6 v^5 w+528 a^8 b^10 c^6 u^6 v^5 w-704 a^6 b^12 c^6 u^6 v^5 w+256 a^4 b^14 c^6 u^6 v^5 w+24 a^2 b^16 c^6 u^6 v^5 w-24 b^18 c^6 u^6 v^5 w+40 a^16 c^8 u^6 v^5 w+288 a^14 b^2 c^8 u^6 v^5 w-672 a^12 b^4 c^8 u^6 v^5 w-416 a^10 b^6 c^8 u^6 v^5 w+1328 a^8 b^8 c^8 u^6 v^5 w-32 a^6 b^10 c^8 u^6 v^5 w-800 a^4 b^12 c^8 u^6 v^5 w+160 a^2 b^14 c^8 u^6 v^5 w+104 b^16 c^8 u^6 v^5 w-64 a^14 c^10 u^6 v^5 w-480 a^12 b^2 c^10 u^6 v^5 w-96 a^10 b^4 c^10 u^6 v^5 w+704 a^8 b^6 c^10 u^6 v^5 w+256 a^6 b^8 c^10 u^6 v^5 w+416 a^4 b^10 c^10 u^6 v^5 w-608 a^2 b^12 c^10 u^6 v^5 w-128 b^14 c^10 u^6 v^5 w+96 a^10 b^2 c^12 u^6 v^5 w+64 a^8 b^4 c^12 u^6 v^5 w-192 a^6 b^6 c^12 u^6 v^5 w-256 a^4 b^8 c^12 u^6 v^5 w+352 a^2 b^10 c^12 u^6 v^5 w-64 b^12 c^12 u^6 v^5 w+112 a^10 c^14 u^6 v^5 w+560 a^8 b^2 c^14 u^6 v^5 w+1216 a^6 b^4 c^14 u^6 v^5 w+1344 a^4 b^6 c^14 u^6 v^5 w+592 a^2 b^8 c^14 u^6 v^5 w+272 b^10 c^14 u^6 v^5 w-112 a^8 c^16 u^6 v^5 w-544 a^6 b^2 c^16 u^6 v^5 w-1056 a^4 b^4 c^16 u^6 v^5 w-672 a^2 b^6 c^16 u^6 v^5 w-176 b^8 c^16 u^6 v^5 w+32 a^4 b^2 c^18 u^6 v^5 w+32 a^2 b^4 c^18 u^6 v^5 w-64 b^6 c^18 u^6 v^5 w+64 a^4 c^20 u^6 v^5 w+160 a^2 b^2 c^20 u^6 v^5 w+128 b^4 c^20 u^6 v^5 w-40 a^2 c^22 u^6 v^5 w-56 b^2 c^22 u^6 v^5 w+8 c^24 u^6 v^5 w+64 a^16 b^2 c^6 u^5 v^6 w-192 a^14 b^4 c^6 u^5 v^6 w+64 a^12 b^6 c^6 u^5 v^6 w+320 a^10 b^8 c^6 u^5 v^6 w-320 a^8 b^10 c^6 u^5 v^6 w-64 a^6 b^12 c^6 u^5 v^6 w+192 a^4 b^14 c^6 u^5 v^6 w-64 a^2 b^16 c^6 u^5 v^6 w+192 a^14 b^2 c^8 u^5 v^6 w+128 a^12 b^4 c^8 u^5 v^6 w-1216 a^10 b^6 c^8 u^5 v^6 w+768 a^8 b^8 c^8 u^5 v^6 w+832 a^6 b^10 c^8 u^5 v^6 w-896 a^4 b^12 c^8 u^5 v^6 w+192 a^2 b^14 c^8 u^5 v^6 w-960 a^12 b^2 c^10 u^5 v^6 w+1216 a^10 b^4 c^10 u^5 v^6 w+1152 a^8 b^6 c^10 u^5 v^6 w-2176 a^6 b^8 c^10 u^5 v^6 w+832 a^4 b^10 c^10 u^5 v^6 w-64 a^2 b^12 c^10 u^5 v^6 w+704 a^10 b^2 c^12 u^5 v^6 w-2304 a^8 b^4 c^12 u^5 v^6 w+1152 a^6 b^6 c^12 u^5 v^6 w+768 a^4 b^8 c^12 u^5 v^6 w-320 a^2 b^10 c^12 u^5 v^6 w+704 a^8 b^2 c^14 u^5 v^6 w+1216 a^6 b^4 c^14 u^5 v^6 w-1216 a^4 b^6 c^14 u^5 v^6 w+320 a^2 b^8 c^14 u^5 v^6 w-960 a^6 b^2 c^16 u^5 v^6 w+128 a^4 b^4 c^16 u^5 v^6 w+64 a^2 b^6 c^16 u^5 v^6 w+192 a^4 b^2 c^18 u^5 v^6 w-192 a^2 b^4 c^18 u^5 v^6 w+64 a^2 b^2 c^20 u^5 v^6 w+8 a^18 b^2 c^4 u^7 v^3 w^2-24 a^16 b^4 c^4 u^7 v^3 w^2-32 a^14 b^6 c^4 u^7 v^3 w^2+224 a^12 b^8 c^4 u^7 v^3 w^2-336 a^10 b^10 c^4 u^7 v^3 w^2+112 a^8 b^12 c^4 u^7 v^3 w^2+224 a^6 b^14 c^4 u^7 v^3 w^2-288 a^4 b^16 c^4 u^7 v^3 w^2+136 a^2 b^18 c^4 u^7 v^3 w^2-24 b^20 c^4 u^7 v^3 w^2-40 a^16 b^2 c^6 u^7 v^3 w^2+96 a^14 b^4 c^6 u^7 v^3 w^2+576 a^12 b^6 c^6 u^7 v^3 w^2-1952 a^10 b^8 c^6 u^7 v^3 w^2+1584 a^8 b^10 c^6 u^7 v^3 w^2+544 a^6 b^12 c^6 u^7 v^3 w^2-1152 a^4 b^14 c^6 u^7 v^3 w^2+288 a^2 b^16 c^6 u^7 v^3 w^2+56 b^18 c^6 u^7 v^3 w^2+64 a^14 b^2 c^8 u^7 v^3 w^2-32 a^12 b^4 c^8 u^7 v^3 w^2-1088 a^10 b^6 c^8 u^7 v^3 w^2+2656 a^8 b^8 c^8 u^7 v^3 w^2-2368 a^6 b^10 c^8 u^7 v^3 w^2+1184 a^4 b^12 c^8 u^7 v^3 w^2-704 a^2 b^14 c^8 u^7 v^3 w^2+288 b^16 c^8 u^7 v^3 w^2-352 a^10 b^4 c^10 u^7 v^3 w^2+96 a^8 b^6 c^10 u^7 v^3 w^2+1088 a^6 b^8 c^10 u^7 v^3 w^2+1728 a^4 b^10 c^10 u^7 v^3 w^2-1248 a^2 b^12 c^10 u^7 v^3 w^2-1312 b^14 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a^6 b^16 c^2 u v^6 w^5-192 a^18 b^2 c^4 u v^6 w^5-128 a^16 b^4 c^4 u v^6 w^5+1216 a^14 b^6 c^4 u v^6 w^5-768 a^12 b^8 c^4 u v^6 w^5-832 a^10 b^10 c^4 u v^6 w^5+896 a^8 b^12 c^4 u v^6 w^5-192 a^6 b^14 c^4 u v^6 w^5+960 a^16 b^2 c^6 u v^6 w^5-1216 a^14 b^4 c^6 u v^6 w^5-1152 a^12 b^6 c^6 u v^6 w^5+2176 a^10 b^8 c^6 u v^6 w^5-832 a^8 b^10 c^6 u v^6 w^5+64 a^6 b^12 c^6 u v^6 w^5-704 a^14 b^2 c^8 u v^6 w^5+2304 a^12 b^4 c^8 u v^6 w^5-1152 a^10 b^6 c^8 u v^6 w^5-768 a^8 b^8 c^8 u v^6 w^5+320 a^6 b^10 c^8 u v^6 w^5-704 a^12 b^2 c^10 u v^6 w^5-1216 a^10 b^4 c^10 u v^6 w^5+1216 a^8 b^6 c^10 u v^6 w^5-320 a^6 b^8 c^10 u v^6 w^5+960 a^10 b^2 c^12 u v^6 w^5-128 a^8 b^4 c^12 u v^6 w^5-64 a^6 b^6 c^12 u v^6 w^5-192 a^8 b^2 c^14 u v^6 w^5+192 a^6 b^4 c^14 u v^6 w^5-64 a^6 b^2 c^16 u v^6 w^5+64 a^14 b^8 c^2 u^6 w^6-128 a^12 b^10 c^2 u^6 w^6-64 a^10 b^12 c^2 u^6 w^6+256 a^8 b^14 c^2 u^6 w^6-64 a^6 b^16 c^2 u^6 w^6-128 a^4 b^18 c^2 u^6 w^6+64 a^2 b^20 c^2 u^6 w^6-128 a^12 b^8 c^4 u^6 w^6+384 a^10 b^10 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
