Overview

Euclidean geometry problems rely on angle chasing and similarity. Power of a point and cyclic quadrilaterals are frequent tools in AMC 12 and AIME.

Key Ideas

• If a quadrilateral is cyclic, opposite angles sum to $180^\circ$.
• Power of a point: for chords through a point, $PA\cdot PB=PC\cdot PD$.
• Similar triangles yield proportional sides and equal angles.

Worked Example

In a circle, chords $AB$ and $CD$ intersect at $P$. If $PA=3$, $PB=5$, and $PC=2$, then $PD=\frac{PA\cdot PB}{PC}=\frac{15}{2}$.

Practice Problems