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: Compare tangent directions

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In plain words

The same complex relation that encodes the metric condition also encodes a quarter-turn between tangent directions.

\ell_1\perp\ell_2

Detailed analysis

In complex coordinates with C=0, A=1, B=b and D=d, the source computes the directions of the two tangents from the circumcenters. Substituting d=(b-bi)/(1-bi) makes the direction quotient equal to -1, hence the tangent lines are perpendicular.