Question 22
(GBR 2; IMO 2) $A,B,C,D$ are four points in the plane, with $C,D$ on the same side of the line $AB$, such that $AC\cdot BD=AD\cdot BC$ and $\angle ADB=90^\circ+\angle ACB$. Find the ratio
$$\frac{AB\cdot CD}{AC\cdot BD},$$
and prove that circles $ACD,BCD$ are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicular.)
$$\frac{AB\cdot CD}{AC\cdot BD},$$
and prove that circles $ACD,BCD$ are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicular.)