Proving with tangents

Most tangent proofs use the same two facts: radius ⊥ tangent (a free 90°) and equal tangents from an outside point.

Pick your tool:

Tangents at the two ends of a diameter are parallel. Both make 90° with the diameter, so the alternate angles are equal.

The angle between two tangents + the angle at the centre = 180°. The four-sided shape OPTQ already has two 90° angles, and its angles add to 360°. If ∠POQ = 110°, then ∠PTQ = 70°.

A quadrilateral drawn round a circle: AB + CD = AD + BC. Each side is made of two tangent pieces. Every corner gives two equal pieces, so both sums contain the same four pieces.

So a parallelogram drawn round a circle must be a rhombus. Opposite sides are equal (it's a parallelogram), and AB + CD = AD + BC then forces neighbouring sides to be equal too.

A triangle drawn round a circle: call the tangent pieces from each corner x, y, z. Each side is the sum of two of them, so three sides give three equations. For sides 8, 10 and 12, the pieces are 3, 5 and 7.

Finding a side from the area: join the centre to the three corners. The big triangle splits into three small ones, each with height r (a radius ⊥ the side). So area = ½ × r × (each side), added up.