Problem 2

In the triangle

ABC

, prove that there is a point

D

on side

AB

such that

CD

is the geometric mean of

AD

and

DB

if and only if

\sin A \sin B \le \sin^2 \frac{C}{2}

Step 1 of 6: Power of the point

D

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In plain words

Turning the algebraic condition $DC^2=DA\cdot DB$ into a purely geometric one, $DC=DE$ , lets us reason about where a point sits on a fixed chord instead of manipulating lengths.

DA\cdot DB = DC\cdot DE

Detailed analysis

Let $D$ be any point on side $AB$ , and extend the line $CD$ to meet the circumcircle of $\triangle ABC$ again at $E$ . By the intersecting-chords relation (power of a point) for the two chords $AB$ and $CE$ meeting at $D$ , we have $DA\cdot DB = DC\cdot DE$ . So $CD$ is the geometric mean of $AD$ and $DB$ , i.e. $DC^2=DA\cdot DB$ , exactly when $DC=DE$ .