Δευτέρα, 19 Δεκεμβρίου 2011

Reflections of AHO,BHO,CHO in the bisectors


Let ABC be a triangle.


Denote:

AabHabOab : = the reflection of AHO in BI

AacHacOac : = the reflection of AHO in CI

Ah : = AabHab /\ AacHac

Ao : = AabOab /\ AacOac

The line AhAo is parallel to BC.

Similarly the lines BhBo, ChCo are parallels to CA, AB, resp.

Which is the homothetic center of the triangles ABC, Triangle bounded by (AhAo, BhBo, ChCo)?

Generalization:
P,P* := two isogonal conjugate points (instead of H,O).
Which is the locus of P such that ABC, Triangle bounded by (ApAp*, BpBp*, CpCp*) are perspective?

APH, 19 December 2011

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For H and O (or O and H, which is the same), the perspector is X318.

For P and P' = isogonal conjugate of P, the result is true.

The perspector is:

{a (a - b - c) v (b^2 u (u + v) + v (-c^2 u + a^2 (u + v))) w (c^2 u (u + w) + w (-b^2 u + a^2 (u + w))),
b (a - b + c) u (b^2 u (u + v) + v (-c^2 u + a^2 (u + v))) w (-c^2 v (v + w) - w (-a^2 v + b^2 (v + w))), (a + b - c) c u v (c^2 u (u + w) + w (-b^2 u + a^2 (u + w))) (-c^2 v (v + w) - w (-a^2 v + b^2 (v + w)))}

For P = (u:v:w) and P' = (x:y:z) the locus is:

(b c u x + a c v y + a b w z) (-a^2 c^2 u v x y +
b^2 c^2 u v x y + a c^3 u v x y - b c^3 u v x y +
a^2 b^2 u w x z - a b^3 u w x z + b^3 c u w x z -
b^2 c^2 u w x z + a^3 b v w y z - a^2 b^2 v w y z -
a^3 c v w y z + a^2 c^2 v w y z) (a b c u^2 w x^2 y -
b^2 c u^2 w x^2 y - b c^2 u^2 w x^2 y - a^2 c u v w x^2 y +
2 a b c u v w x^2 y - b^2 c u v w x^2 y - b c^2 u v w x^2 y +
c^3 u v w x^2 y - a^2 c v^2 w x^2 y + a b c v^2 w x^2 y +
a b c u w^2 x^2 y - b c^2 u w^2 x^2 y + a b c v w^2 x^2 y +
a b c u^2 w x y^2 - b^2 c u^2 w x y^2 - a^2 c u v w x y^2 +
2 a b c u v w x y^2 - b^2 c u v w x y^2 - a c^2 u v w x y^2 +
c^3 u v w x y^2 - a^2 c v^2 w x y^2 + a b c v^2 w x y^2 -
a c^2 v^2 w x y^2 + a b c u w^2 x y^2 + a b c v w^2 x y^2 -
a c^2 v w^2 x y^2 + a b c u^2 v x^2 z - b^2 c u^2 v x^2 z -
b c^2 u^2 v x^2 z + a b c u v^2 x^2 z - b^2 c u v^2 x^2 z -
a^2 b u v w x^2 z + b^3 u v w x^2 z + 2 a b c u v w x^2 z -
b^2 c u v w x^2 z - b c^2 u v w x^2 z + a b c v^2 w x^2 z -
a^2 b v w^2 x^2 z + a b c v w^2 x^2 z - a^2 c u^2 v x y z +
2 a b c u^2 v x y z - b^2 c u^2 v x y z - b c^2 u^2 v x y z +
c^3 u^2 v x y z - a^2 c u v^2 x y z + 2 a b c u v^2 x y z -
b^2 c u v^2 x y z - a c^2 u v^2 x y z + c^3 u v^2 x y z -
a^2 b u^2 w x y z + b^3 u^2 w x y z + 2 a b c u^2 w x y z -
b^2 c u^2 w x y z - b c^2 u^2 w x y z + 2 a^3 u v w x y z -
2 a^2 b u v w x y z - 2 a b^2 u v w x y z + 2 b^3 u v w x y z -
2 a^2 c u v w x y z + 6 a b c u v w x y z - 2 b^2 c u v w x y z -
2 a c^2 u v w x y z - 2 b c^2 u v w x y z + 2 c^3 u v w x y z +
a^3 v^2 w x y z - a b^2 v^2 w x y z - a^2 c v^2 w x y z +
2 a b c v^2 w x y z - a c^2 v^2 w x y z - a^2 b u w^2 x y z -
a b^2 u w^2 x y z + b^3 u w^2 x y z + 2 a b c u w^2 x y z -
b c^2 u w^2 x y z + a^3 v w^2 x y z - a^2 b v w^2 x y z -
a b^2 v w^2 x y z + 2 a b c v w^2 x y z - a c^2 v w^2 x y z -
a^2 c u^2 v y^2 z + a b c u^2 v y^2 z - a^2 c u v^2 y^2 z +
a b c u v^2 y^2 z - a c^2 u v^2 y^2 z + a b c u^2 w y^2 z +
a^3 u v w y^2 z - a b^2 u v w y^2 z - a^2 c u v w y^2 z +
2 a b c u v w y^2 z - a c^2 u v w y^2 z - a b^2 u w^2 y^2 z +
a b c u w^2 y^2 z + a b c u^2 v x z^2 - b c^2 u^2 v x z^2 +
a b c u v^2 x z^2 - a^2 b u v w x z^2 - a b^2 u v w x z^2 +
b^3 u v w x z^2 + 2 a b c u v w x z^2 - b c^2 u v w x z^2 -
a b^2 v^2 w x z^2 + a b c v^2 w x z^2 - a^2 b v w^2 x z^2 -
a b^2 v w^2 x z^2 + a b c v w^2 x z^2 + a b c u^2 v y z^2 +
a b c u v^2 y z^2 - a c^2 u v^2 y z^2 - a^2 b u^2 w y z^2 +
a b c u^2 w y z^2 + a^3 u v w y z^2 - a^2 b u v w y z^2 -
a b^2 u v w y z^2 + 2 a b c u v w y z^2 - a c^2 u v w y z^2 -
a^2 b u w^2 y z^2 - a b^2 u w^2 y z^2 + a b c u w^2 y z^2) == 0

hence we have a line, a conic and a cubic.

A particular case:

For P = H, the locus for P'=(x:y:z) factors as:

* Trilinear polar of isogonal conjugate of X2183
* Trilinear polar of X63
* Circumcircle
* Circumhyperbola through O and I.

Francisco Javier García Capitán
19 December 2011

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