Let
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
,
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
,
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
,
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
be four points in the plane, with
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
and
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
on the same side of the line
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
, such that
![AC \cdot BD = AD \cdot BC](/media/m/c/a/4/ca4e677eb0bf46cfe2cb310a03298965.png)
and
![\angle ADB = 90^{\circ}+\angle ACB](/media/m/a/a/3/aa3af2cbc21f86bcd8318f70c366f4eb.png)
. Find the ratio
and prove that the circumcircles of the triangles
![ACD](/media/m/0/b/1/0b171034d79122bd02f64bc8f6ae94dd.png)
and
![BCD](/media/m/3/e/e/3eefa3e34f78e628cbb5cd3988774661.png)
are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicuar. Thus, proving that the circumcircles of the triangles
![ACD](/media/m/0/b/1/0b171034d79122bd02f64bc8f6ae94dd.png)
and
![BCD](/media/m/3/e/e/3eefa3e34f78e628cbb5cd3988774661.png)
are orthogonal is equivalent to proving that the tangents to the circumcircles of the triangles
![ACD](/media/m/0/b/1/0b171034d79122bd02f64bc8f6ae94dd.png)
and
![BCD](/media/m/3/e/e/3eefa3e34f78e628cbb5cd3988774661.png)
at the point
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
are perpendicular.)
%V0
Let $A$, $B$, $C$, $D$ be four points in the plane, with $C$ and $D$ 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 the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicuar. Thus, proving that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal is equivalent to proving that the tangents to the circumcircles of the triangles $ACD$ and $BCD$ at the point $C$ are perpendicular.)