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: Encode both hypotheses

Previous step Next step

In plain words

A complex quotient records a ratio and an angle at once.

\frac{b-d}{1-d}=bi

Detailed analysis

The length equality gives the modulus of (b-d)/(1-d), while the angle condition gives its argument as the direction of b followed by a quarter-turn. Therefore the quotient equals bi.