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 2 of 4: Transform the length condition

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In plain words

The multiplicative condition is designed to become equality of two image lengths.

AC\cdot BD=AD\cdot BC\Longrightarrow B'D'=B'C

Detailed analysis

Use the inversion distance formula for AC, BD, AD and BC. Substitution into the given product equality cancels the common radius factors and yields B'D'=B'C.