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 4 of 4: Prove perpendicular tangents

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In plain words

The same quarter-turn encoded by the hypotheses appears as a minus-one quotient between tangent directions.

\ell_1\perp\ell_2

Detailed analysis

For the tangents at C to the circles ACD and BCD, use the normalized complex coordinates from the third approach. The first similarity gives the same expression for d as in that approach; the circumcenter directions of the two circles then have quotient -1. Hence the two tangent directions at C are perpendicular.