Problem 2

Let

D

be a point inside acute triangle

ABC

such that

\angle ADB=\angle ACB+\frac{\pi}{2}

and

AC\cdot BD=AD\cdot BC

. (a) Calculate the ratio

\frac{AB\cdot CD}{AC\cdot BD}

. (b) Prove that the tangents at

C

to the circumcircles of

\triangle ACD

and

\triangle BCD

are perpendicular.

Step 1 of 4: Normalize the plane

In plain words

A similarity normalization removes irrelevant scale and position.

C=0,\ A=1,\ B=b,\ D=d

Detailed analysis

Represent points by complex numbers and choose the similarity so that C=0 and A=1. Write B=b and D=d.