## Tangential Distances

#### Problem

$P$ represents a remote point from a circle, and the lines through $PS$ and $PT$ are tangential to the circle.

Prove that the tangential distances $PS$ and $PT$ are always equal.

#### Solution

Consider the following diagram where $C$ represents the centre of the circle.

As a radius meets a tangent at a right angle, $PCS$ and $PCT$ are right angled triangles.

By the Pythagorean Theorem, $(PS)^2 = (PC)^2 - (CS)^2$ and $(PT)^2 = (PC)^2 - (CT)^2$. But as $CS = CT$ (they are both radii), it follows that $(PS)^2 = (PT)^2 \Rightarrow PS = PT$. **Q.E.D.**

Problem ID: 351 (17 Apr 2009) Difficulty: 2 Star