Tangents and Chords

A tangent touches the circle once, and meets the radius at that point at exactly 90°.

Two tangents from a point

The two tangent lengths from any external point are equal.

Chords

A radius perpendicular to a chord cuts it exactly in half.

Why the right angle matters

It turns most tangent problems into Pythagoras problems.

Tangent meets radius at a right angle

A tangent touches the circle at exactly one point, and the radius drawn to that point is perpendicular to it. Any other angle would let the line cross the circle twice, contradicting tangency.

Two tangents from a point are equal

From an external point, the two tangent segments to a circle have equal length. The proof pairs two right triangles sharing a hypotenuse and a radius, congruent by RHS.

Chords and their perpendicular bisectors

The perpendicular from the centre to a chord bisects it, and the perpendicular bisector of any chord passes through the centre. This is how you locate the centre of a circle from an arc, which surveyors and machinists actually do.

Draw the radii

The standard first move in almost any circle problem is to draw radii to every marked point. They are all equal, which creates isosceles triangles, which give equal angles. Many circle proofs are only that observation applied.