Question 1
Let $ABC$ be a triangle that is not right-angled at $A$, and let $AM$ be its median. A point $D$ varies on line $AM$. Let $(O_1)$ and $(O_2)$ be circles passing through $D$ and tangent to $BC$ at $B$ and $C$, respectively. Let $P$ and $Q$ be the intersections of line $AB$ with $(O_1)$ and line $AC$ with $(O_2)$, respectively. Prove that:
1. The tangents to $(O_1)$ at $P$ and to $(O_2)$ at $Q$ meet at a point $S$.
2. As $D$ varies on $AM$, the point $S$ lies on a fixed line.
Interpretation needed for the printed construction: $D\ne M$, so the two circles exist; $P$ and $Q$ mean the second intersections other than $B$ and $C$.
1. The tangents to $(O_1)$ at $P$ and to $(O_2)$ at $Q$ meet at a point $S$.
2. As $D$ varies on $AM$, the point $S$ lies on a fixed line.
Interpretation needed for the printed construction: $D\ne M$, so the two circles exist; $P$ and $Q$ mean the second intersections other than $B$ and $C$.