Question 132
Let $A$ and $C$ be two points on a circle $C$ such that $(AC)$ is not a diameter, and let $P$ be a point of the line segment $(AC)$, other than its midpoint. Circles $c_1$ and $c_2$ are interiorly tangent to the circle $C$ at $A$ and $C$, respectively. They both pass through $P$ and intersect again at $Q$. The line $PQ$ intersects circle $C$ at $B$ and $D$. Circle $c_1$ intersects line segments $AB$ and $AD$ at $K$ and $N$, respectively, while circle $c_2$ intersects line segments $CB$ and $CD$ at $L$ and $M$, respectively. Prove that:
a) the quadrilateral $KLMN$ is an isosceles trapezoid;
b) $Q$ is the midpoint of the line segment $BD$.
a) the quadrilateral $KLMN$ is an isosceles trapezoid;
b) $Q$ is the midpoint of the line segment $BD$.