Construction of Tangent from Point to Circle/Proof 2

Theorem
From a given point outside a given circle, it is possible to draw a tangent to that circle.

Proof

 * Euclid-III-17a.png

Let $BCD$ with center $A$ be the circle, and let $E$ be the exterior point from which a tangent is to be drawn.

Bisect $AE$ at $F$.

Then draw a circle $AEG$ whose center is $F$ and whose radius is $AF$.

The point $G$ is where $AEG$ intersects $BCD$.

The line $EG$ is the required tangent.

Proof of Construction
By the method of construction, $AE$ is the diameter of $AEG$.

By Thales' Theorem $\angle AGE$ is a right angle.

But $AG$ is a radius of $BCD$.

The result follows from Radius at Right Angle to Tangent.