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 3 of 4: Find D and part (a)

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In plain words

The metric ratio is independent of the particular triangle after normalization.

d=\frac{b-bi}{1-bi}

Detailed analysis

Solve the quotient equation for d. Substitution into the requested ratio cancels the common factors and leaves the modulus of 1-i, namely the square root of 2.