CONCYCLIC CIRCUMCENTERS - 2

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

Denote:

Ab,Ac = the reflections of A' in CC',BB', resp. (lying on AC, AB, resp.)

Similarly Bc,Ba = the reflections of B' in AA',CC', resp. and Ca,Cb = the reflections of C' in BB',AA', resp.

A1B1C1 = the antipedal triangle of I wrt triangle A'AbAc

Similarly A2B2C2 = the antipedal triangle of I wrt triangle B'BcBa and A3B3C3 = the antipedal triangle of I wrt triangle C'CaCb,

Oa,Ob,Oc = the circumcenters of the triangles A1B1C1, A2B2C2, A3B3C3, resp.

The circumcenter O of ABC and the circumcenters Oa,Ob,Oc of A1B1C1,A2B2C2,A3B3C3, resp. are concyclic.

Which point is the center X of the circle?

Antreas P. Hatzipolakis, 4 June 2014

X is X(5495)

Peter Moses 5 June 2014

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 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

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

POINT ON THE CIRCUMCIRCLE

Let ABC be a triangle, A'B'C', A"B"C" the antipedal tiangles of H, O [antimedial, tangential, resp.].

The triangles:

3. ABC, Triangle A*B*C* bounded by the lines (A'A",B'B",C'C") are perspective.

[the perspector is on the circumcircle(ABC) = NPC(A'B'C') = Incircle(A"B"C")]

Coordinates?

Antreas P. Hatzipolakis, Hyacinthos #21780

Generalization

CIRCUMCIRCLE

Let ABC be a triangle and P, Q two isogonal conjugate points and P1P2P3, Q1Q2Q3 the pedal triangles of P,Q, resp.

Denote:

R1 = the radical axis of the circles ((P1, P1Q)[=circle centered at P1 with radius P1Q],(Q1, Q1P))

R2 = the radical axis of the circles ((P2, P2Q),(Q2, Q2P))

R3 = the radical axis of the circles ((P3, P3Q),(Q3, Q3P))

The triangles ABC, Triangle A'B'C' bounded by (R1,R2,R3) are perspective at a point D on the circumcircle.

Antreas P. Hatzipolakis, Hyacinthos #21733

EULER LINES, CIRCUMCIRCLES

Let ABC be a triangle, L,La,Lb,Lc the Euler lines of ABC, IBC, ICA, IAB, resp. (concurrent at S) and Oa,Ob,Oc the circumcenters of IBC, ICA, IAB, resp.

Denote:

Ab, Ac = the orthogonal projections of Oa on Lb,Lc, resp.

Bc, Ba = the orthogonal projections of Ob on Lc,La, resp.

Ca, Cb = the orthogonal projections of Oc on La,Lb, resp.

O1, O2, O3 = the circumcenters of OaAbAc, ObBcBa, OcCaCb, resp.

1. The triangles ABC, O1O2O3 are perspective.

2. The circumcenter of O1O2O3 is on the line L.

Antreas P. Hatzipolakis, 8 Febr. 2013

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

1. The triangles ABC, O1O2O3 are perspective.

Yes at

Q = a (2 a^3-2 a^2 b-2 a b^2+2 b^3-a^2 c-a b c-b^2 c-2 a c^2-2 b c^2+c^3) (2 a^3-a^2 b-2 a b^2+b^3-2 a^2 c-a b c-2 b^2 c-2 a c^2-b c^2+2 c^3)::

ETC Search = 2.1298183373048648210

Q is on lines {{21,4867},{79,2646},{80,3584},{758,2320},{1389,3746}} and on the Feuerbach hyperbola.

Q = 6 (r+R) X[21] +(2 r-R) X[4867]

Q = R X[79] + 4 (2 r+R) X[2646]

The isogonal of Q, gQ, is on lines {{1,3},{2,4867},{8,3841},{21,4084},{79,3671},{80,226},{81,759},{100,3919},{191,4018},{515,3982},{519,5249},{758,3219},{956,3894},{958,3901},{993,4880},{1001,3899},{1100,5341},{1203,3924},{1411,2003},{2802,3957},{3585,3649},{3636,5330},{3868,5258},{3869,5259},{3874,5288},{3881,4861},{3918,4420},{4067,5260}};

gQ = (4 r + 5 R)X[1] - 2 r X[3]

gQ = (2 r + 3 R) X[21] + 2 R X[4084]

ETC Search = 1.2648627122986571860

O1O2O3 is also perspective to various other triangles too; for example:

1. Excentral at

a (3 a^3-a^2 b-3 a b^2+b^3-a^2 c-3 a b c-3 b^2 c-3 a c^2-3 b c^2+c^3) :: = X[1] + 2 X[21]

on lines {{1,21},{30,1699},{35,3753},{36,3742},{100,3968},{210,5251},{214,5284},{442,3586},{484,3919},{1125,2475},{1420,3649},{1698,1837},{2320,3065},{2646,5259},{3158,3679},{3219,4525},{3336,4189},{3337,5267},{3616,4299},{3636,3648},{3683,4867},{3746,3880},{3956,5260},{4316,5249},{4539,5302},{4677,4933}}. ETC search = 2.6815474100289784017

2. Intouch at

a (a+b-c) (a-b+c) (2 a^4-2 a^3 b-2 a^2 b^2+2 a b^3-2 a^3 c-2 a^2 b c+b^3 c-2 a^2 c^2+2 b^2 c^2+2 a c^3+b c^3) :: R (r+4 R) X[7] - r (4 r+7 R) X[21]

on lines {{7,21},{11,30},{12,5251},{100,5172},{191,1420},{392,3647},{758,1319},{1317,2078},{1411,1758},{2771,5126},{3651,5204},{4189,5221}}. ETC search -1.0652570342486130228

2. The circumcenter of O1O2O3 is on the line L.

Yes, at

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

on lines {{2,3},{36,3649},{191,3576},{214,960},{758,1385},{952,5258},{1837,5010},{3579,3754},{3650,5303}}

also

P = 3 R X[2] + (4 r+3 R) X[3]

P = (2 r - R) X[36] + R X[3649]

P = X[191] + 3 X[3576]

P = (2 r +R) X[3579] + 2 R X[3754]

P = R X[3650] - (6 r+R) X[5303]

ETC Search = 4.9797648772556482798.

Peter J. C. Moses, 8 Febr. 2013

ETC X5424, X5426, X5427, X5428

SQUARE PROBLEM

Let ABCD be a square. E is a variable point on the line BC, I is the midpoint of BE. CD and AE meet at F, IF and DE meet at M. The locus of M is the circumcircle of the square ABCD.

Bernard Gibert, Hyacinthos Message #19822

Solution:

To prove that M lies on the circumcircle of ABCD is enough to prove that the quadrilateral BMCD is cyclic, and since DCB = 90 d., to prove that EMB = 90 d. If triangle EMB is right angled, then since IB = IE ==> IB = IE = IM.

I will prove that IE = IM.

I take the point E between B,C (if E is on the extension of BC, then signs in the calculations are changed).

Denote AB = BC = CD = DA := a and EC := x

We have:

IE = (a-x)/2

From the similar triangles FAD,FEC we get:

CF = ax / (a-x)

DF = a^2 / (a-x)

In the right triangle ICF we have:

IF^2 = IC^2 + CF^2 = (IE + EC)^2 + CF^2 = (a^2 + x^2) / 2(a-x)

Now, by Menelaus Theorem in the triangle ICF with transversal MED, we get:

MI/MF . DF/DC . EC/EI = 1

==>

[IM /(IF-IM)]. DF/DC . EC/EI = 1

==>

(IM / [[(a^2+x^2) / 2(a-x)] - IM]) . (a^2 / (a-x)]/a) . (x /[(a-x)/2]) = 1

==> IM = (a-x)/2

So IE = IM,

QED

Antreas