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: Obtain a second similarity

Previous step Next step

In plain words

Two similarities transfer the unknown CD into the easily measured auxiliary triangle.

\triangle ABB'\sim\triangle ADC

Detailed analysis

Directed-angle comparison gives angle CAB' equal to DAB, hence angle CAD equal to BAB'. The length ratio from the first similarity and AC/AD then gives triangle ABB' similar to ADC.