GULF 2019 Problem 1

Let $ABCD$ be a trapezium (trapezoid) with $AD$ parallel to $BC$ and $J$ be the intersection of the diagonals $AC$ and $BD$ . Point $P$ a chosen on the side $BC$ such that the distance from $C$ to the line $AP$ is equal to the distance from $B$ to the line $DP$ .

The following three questions 1, 2 and 3 are independent, so that a condition in one question does not apply in another question.

1.Suppose that $Area( \vartriangle AJB) =6$ and that $Area(\vartriangle BJC) = 9$ . Determine $Area(\vartriangle APD)$ . 2. Find all points $Q$ on the plane of the trapezium such that $Area(\vartriangle AQB) = Area(\vartriangle DQC)$ . 3. Prove that $PJ$ is the angle bisector of $\angle APD$ .