Overview

Homothety and spiral similarity explain many circle tangency and ratio problems. They are powerful for advanced triangle configurations.

Key Ideas

• Homothety centered at $O$ maps $P$ to $P'$ with $\vec{OP'} = k\vec{OP}$.
• Spiral similarity combines rotation and dilation.
• Parallel lines are preserved under homothety.

Core Skills

Locate the Center

The homothety center lies on lines joining corresponding points. Use two pairs to locate it.

Use Spiral Similarity

If two segments subtend equal angles, there is a spiral similarity mapping one to the other. Use it to relate lengths and angles.

Track Scale Factors

Lengths scale by $k$, areas by $k^2$. Apply this to ratios quickly.

Worked Example

If a homothety with ratio $2$ maps segment $AB$ to $A'B'$, and $AB=5$, find $A'B'$.

Lengths scale by $2$, so $A'B'=10$.

More Examples

Example 1: Area Scaling

If a homothety has ratio $3$, by what factor do areas scale?

$9$.

Example 2: Center on Lines

If $A\to A'$ and $B\to B'$ under a homothety, the center lies on $AA'$ and $BB'$.

Use their intersection to locate the center.

Example 3: Spiral Similarity Angle

If $\angle ABC = \angle ADE$, then there is a spiral similarity sending $BC$ to $DE$.

Strategy Checklist

• Use two pairs of corresponding points to find a homothety center.
• Apply scale factors to lengths and areas.
• Use spiral similarity to match angle pairs.

Common Pitfalls

• Confusing homothety with reflection.
• Forgetting that areas scale by $k^2$.
• Mixing up the direction of the scale factor.

Practice Problems