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.

Core Skills

Angle Chasing

Mark known angles, use linear pairs and triangle sums, and track equal angles from parallel lines or cyclic quadrilaterals.

Similarity First

Look for AA similarity before heavy computation. Similar triangles give proportions and angle equality quickly.

Use Power of a Point

Translate secant and chord configurations into product equalities.

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 $PD=\frac{PA\cdot PB}{PC}=\frac{15}{2}$.

More Examples

Example 1: Cyclic Angle

If $ABCD$ is cyclic and $\angle A=70^\circ$, find $\angle C$.

$110^\circ$.

Example 2: Similar Triangles

If $\triangle ABC \sim \triangle DEF$ and $AB/DE=3/2$, find the area ratio.

The area ratio is $(3/2)^2=9/4$.

Example 3: Power of a Point

From external point $P$, a secant meets the circle at $A,B$ with $PA=4$ and $PB=9$. Find the tangent length.

$PT^2=4\cdot 9=36$, so $PT=6$.

Strategy Checklist

• Mark angles and look for cyclic quadrilaterals.
• Seek similar triangles before computing lengths.
• Translate circle configurations into power-of-point equations.

Common Pitfalls

• Assuming cyclicity without proof.
• Using a similarity ratio for areas without squaring.
• Confusing tangent and secant lengths.

Practice Problems