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: Transform the angle

In plain words

Inversion turns the original quarter-turn condition into a visible right angle.

\angle C'B'D'=90^{\circ}

Detailed analysis

Invert about A. Inversion preserves the relevant directed angles, so the two given angle relations combine to make C'B'D' a right angle.