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: Construct the auxiliary point

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In plain words

The right angle packages the added quarter-turn in the hypothesis.

BC=B'C,\quad\angle BCB'=90^{\circ}

Detailed analysis

Choose B' on the same side of AC as B with BC=B'C and angle BCB' a right angle. The given angle and length conditions make triangles ADB and ACB' similar.