Question 43
On the plane given a circle $(O)$ and a point $A$ lying outside the circle. Through $A$, draw the tangents to $(O)$; let $B$ and $C$ be the tangent points. Consider a point $P$ moving on the opposite ray to ray $BA$ and a point $Q$ moving on the opposite ray to ray $CA$, such that the line $PQ$ is tangent to $(O)$. The line $BC$ intersects the line passing through $P$, parallel with $AC$, in $E$, and intersects the line passing through $Q$, parallel to $AB$, in $F$. Show that
1/ The line $EQ$ passes through a fixed point, say, $M$; the line $FP$ passes through a fixed point, say, $N$.
2/ The product $|PM| \cdot |QN|$ is constant.
1/ The line $EQ$ passes through a fixed point, say, $M$; the line $FP$ passes through a fixed point, say, $N$.
2/ The product $|PM| \cdot |QN|$ is constant.