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 the two tangents

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In plain words

A quarter-turn is represented algebraically by multiplication by -1 in the tangent direction quotient.

\frac{z}{\overline z}=-\frac{o_1}{\overline{o_1}}

Detailed analysis

For a circle through the origin, the tangent at the origin is perpendicular to its circumcenter vector. Compute the two circumcenters and substitute the formula for d; the resulting direction ratios differ by -1, proving perpendicularity.