Reflection Method

A and B are two fixed points. Line l is fixed. Point P lies on l and can move. Find min{PA+PB}.

Approach: Reflect point A about l to get A'.

By Toolkit 88 — Triangle Inequality. Triangle Inequality in $\triangle A'BP$:

⇒ Q is the point of interest.

Suppose ray AP reflect and goes to Q in square ABCD.

We can reflect the squares and find corresponding intersection points.