Does it have to be a right angle?

Question

Say you have a circle $O$ and a point on the circle $P$. From P, we create 2 points $A$ and $B$ on the circle such that $PA=X$, $PB=Y$, and the 2 points are on different sides of $\overline{PO}$ (meaning that exactly one of the points is in each of the 2 regions of the plane created by $\overline{PO}$). The diameter of circle $O$ is $Z$, and $X^2+Y^2=Z^2$. Does $\angle APB$ have to be right? More specifically, does $\overline{AB}$ have to be a diameter?

Answer 1

Yes. Construct point D that is the other intersection of the line through AO with the circle. Then $X^2+PD^2=Z^2$, so that $PD^2 = Y^2$ and then $PD=Y$. Since $D$ and $B$ are on the same side of $PO$, they are the same point.