HYACINTHOS 28838

[Kadir Altintas]

Let ABC be a triangle and DEF, MaMbMc the pedal triangles of H,O, resp.

Denote:

(wa) = the circle passing through D, Ma and tangent internally the circumcircle on the positive side of BC
Similarly (wb), (wc)

The circle (w) tangent internally to (wa), (wb), (wc) is tangent to NPC of ABC.

Touchpoint and Center of the circle ?

 

[Peter Moses]:

Hi Antreas,

X(20033) on lines {{2,20032},{427,20034}}.

Center of circle (w) is

(Sin[B] Tan[A] Sqrt[Tan[B]]-Sin[A] Sqrt[Tan[A]] Tan[B]) (Sin[2 A] Sin[C] (Sin[C]-Sqrt[Tan[A] Tan[B]]) Tan[C]-Sin[A] Sin[2 C] Tan[A] (Sin[A]-Sqrt[Tan[B] Tan[C]]))-
(Sin[C] Tan[A] Sqrt[Tan[C]]-Sin[A] Sqrt[Tan[A]] Tan[C]) (Sin[2 A] Sin[B] (Sin[B]-Sqrt[Tan[A] Tan[C]]) Tan[B]-Sin[A] Sin[2 B] Tan[A] (Sin[A]-Sqrt[Tan[B] Tan[C]])) : :

= a SB SC (4 a (b^2-c^2) S^2 SA+b (a^6-2 a^4 b^2+a^2 b^4-5 a^4 c^2+b^4 c^2+5 a^2 c^4-c^6) Sqrt[SA SB]-c (a^6-5 a^4 b^2+5 a^2 b^4-b^6-2 a^4 c^2+a^2 c^4+b^2 c^4) Sqrt[SA SC]-4 a b c (b^2-c^2) SA Sqrt[SB SC])::

on lines {{3,20032},{4,20034},{5,20033}} and on K742.  

(assuming "positive side of BC" means the side of the triangle not containing the incenter.)

See attached pic.

Best regards,
Peter Moses.