The Intersecting Chords Theorem asserts the following:

Given a point P in the interior of a circle, pass two lines through P that intersect the circle in points A and D and, respectively, B and C. Then

The proof follows easily from the similarity of triangles ABP and CDP that is a consequence of the equality of their angles:

as inscribed angles subtended by the same chord BD, | ||

| as inscribed angles subtended by the same chord AC, | |

| as a pair of vertical angles. |

(*Vertical angles* are formed by the same intersecting lines, but the opposite rays - the halves of the lines - cut off by their point of intersection.)

From the similarity of triangles ABP and CDP, we obtain the proportion:

(*) | AP/CP = BP/DP = AB/CD. |

The first identity (AP/CP = BP/DP) leads directly to the Intersecting Chords Theorem: