Question 171
Given an acute, non-isosceles triangle $ABC$, and a point $P$ inside the triangle such that $\angle APB = \angle APC = \alpha$ with $\alpha \gt 80^\circ - \angle BAC$. The circle $(APB)$ intersects the line $AC$ at $E$, the circle $(APC)$ intersects the line $AB$ at $F$. Let $Q$ be the inside point of the triangle $AEF$ such that $\angle AQE = \angle AQF = \alpha$. Let $D$ be the symmetric point of $Q$ through $EF$. The bisector of $\angle EDF$ intersects $AP$ at $T$.
a) Prove that $\angle DET = \angle ABC$, $\angle DFT = \angle ACB$.
b) The line $AP$ intersects the line $DE$, $DF$ respectively at $M$, $N$. Let $I$ and $J$ be centers of incircles of $PEM$ and $PFN$ respectively. Let $K$ be the center of $(DIJ)$. The line $DT$ intersects $(K)$ at $H$. Prove that $HK$ goes through the incircle center of the triangle $DMN$.
a) Prove that $\angle DET = \angle ABC$, $\angle DFT = \angle ACB$.
b) The line $AP$ intersects the line $DE$, $DF$ respectively at $M$, $N$. Let $I$ and $J$ be centers of incircles of $PEM$ and $PFN$ respectively. Let $K$ be the center of $(DIJ)$. The line $DT$ intersects $(K)$ at $H$. Prove that $HK$ goes through the incircle center of the triangle $DMN$.