Chords, secants and tangents of a circle

Refresher on the circle: a chord joins two points of the circle, a secant is a line through two points of the circle, a tangent touches it in one point, and points on one circle are concyclic. We use one fact: angles on the same side of a chord that look at the same arc are equal. Theorem: if two chords AB and CD (or their extensions) meet at E, inside or outside the circle, then EA × EB = EC × ED. Why: draw AD and CB. The angles ∠EAD and ∠ECB stand on the same chord BD (in the inside case; for E outside one of them is the exterior angle of the cyclic quadrilateral), so they are equal, and the angle at E is common or vertical. So △EAD ~ △ECB, which gives EA/EC = ED/EB, that is EA × EB = EC × ED. Always measure from E to each end: for a point E outside, EB is the whole secant, not AB.