Proof Length of Tangent Squared Equals Product of Chord With Chord Extended to Meet Tangent

Theorem

The length of a tangent squared equals product of chord with chord extended to meet the tangent.

Proof

Draw chords AD and BD.

Now consider the trianglesand

Angleis common to both triangles, so call itCall anglethen other angles can be found in terms ofand

Now draw radii fromto the centre of the circle and label the remaining angles.

and

Hence trianglesare similar.

Hence