Question 168
Let $(O)$ be a circle, and $BC$ be a chord of $(O)$ such that $BC$ is not a diameter. Let $A$ be a point on the larger arc $BC$ of $(O)$, and let $E$ and $F$ be the feet of the perpendiculars from $B$ and $C$ to $AC$, $AB$ respectively.
1. Prove that the two tangents of $(AEF)$ at $E$ and $F$ intersect at a fixed point $M$ when $A$ moves on the larger arc $BC$ of $(O)$.
2. Let $T$ be the intersection of $EF$ and $BC$, and $H$ be the orthocenter of $ABC$. Prove that $TH$ is perpendicular to $AM$.
1. Prove that the two tangents of $(AEF)$ at $E$ and $F$ intersect at a fixed point $M$ when $A$ moves on the larger arc $BC$ of $(O)$.
2. Let $T$ be the intersection of $EF$ and $BC$, and $H$ be the orthocenter of $ABC$. Prove that $TH$ is perpendicular to $AM$.