Question 164
Let $ABC$ be a triangle with circumcircle $(O)$, incircle $(I)$, and excircle $(J)$ with respect to vertex $A$. Consider $D, E, F$ to be the tangent points of $(J)$ with $BC, CA, AB$, respectively.
a. Let $L$ be the midpoint of $BC$. The circle with diameter $LJ$ intersects $DE, DF$ again at $K, H$, respectively. Prove that the circles $(BDK)$ and $(CDH)$ meet again at a point on circle $(J)$.
b. Assume that $EF$ intersects $BC$ at $G$. Let $M, N$ be the intersections of $GJ$ with $AB, AC$, respectively. Consider $P, Q$ on $JB, JC$ respectively such that $\angle PAB = \angle QAC = 90^\circ$. Denote $T$ as the intersection of $PM, QN$ and $S$ as the midpoint of the major arc $BC$ of $(O)$. Prove that $SI, AT$ intersect at a point on the circle $(O)$.
a. Let $L$ be the midpoint of $BC$. The circle with diameter $LJ$ intersects $DE, DF$ again at $K, H$, respectively. Prove that the circles $(BDK)$ and $(CDH)$ meet again at a point on circle $(J)$.
b. Assume that $EF$ intersects $BC$ at $G$. Let $M, N$ be the intersections of $GJ$ with $AB, AC$, respectively. Consider $P, Q$ on $JB, JC$ respectively such that $\angle PAB = \angle QAC = 90^\circ$. Denote $T$ as the intersection of $PM, QN$ and $S$ as the midpoint of the major arc $BC$ of $(O)$. Prove that $SI, AT$ intersect at a point on the circle $(O)$.