GULF 2014 Problem 3

(i) $ABC$ is a triangle with a right angle at $A$ , and $P$ is a point on the hypotenuse $BC$ . The line $AP$ produced beyond $P$ meets the line through $B$ which is perpendicular to $BC$ at $U$ . Prove that $BU = BA$ if, and only if, $CP = CA$ . (ii) $A$ is a point on the semicircle $CB$ , and points $X$ and $Y$ are on the line segment $BC$ . The line $AX$ , produced beyond $X$ , meets the line through $B$ which is perpendicular to $BC$ at $U$ . Also the line $AY$ , produced beyond $Y$ , meets the line through $C$ which is perpendicular to $BC$ at $V$ . Given that $BY = BA$ and $CX = CA$ , determine the angle $\angle VAU$ .