ILL 1987 Problem 5

Let there be given three circles $K_1,K_2,K_3$ with centers $O_1,O_2,O_3$ respectively, which meet at a common point $P$ . Also, let $K_1 \cap K_2 = \{P,A\}, K_2 \cap K_3 = \{P,B\}, K_3 \cap K_1 = \{P,C\}$ . Given an arbitrary point $X$ on $K_1$ , join $X$ to $A$ to meet $K_2$ again in $Y$ , and join $X$ to $C$ to meet $K_3$ again in $Z.$ (a) Show that the points $Z,B, Y$ are collinear.(b) Show that the area of triangle $XY Z$ is less than or equal to $4$ times the area of triangle $O_1O_2O_3.$