Question 4
Let $ABC$ be a triangle.
For a given point $P$ of $(BC)$, denote by $E(P)$ and $F(P)$ the second intersection points of the lines $(AB)$ and $(AC)$ with the circle of diameter $[AP]$. Let $T(P)$ be the intersection of the tangents to this circle at $E(P)$ and $F(P)$.
Prove that as $P$ varies on $(BC)$, the locus of $T(P)$ is a line.
For a given point $P$ of $(BC)$, denote by $E(P)$ and $F(P)$ the second intersection points of the lines $(AB)$ and $(AC)$ with the circle of diameter $[AP]$. Let $T(P)$ be the intersection of the tangents to this circle at $E(P)$ and $F(P)$.
Prove that as $P$ varies on $(BC)$, the locus of $T(P)$ is a line.