The Intersecting Chords Theorem asserts the following:
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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:
 BAD = BCD, | as inscribed angles subtended by the same chord BD, | |
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ABC = ADC, | as inscribed angles subtended by the same chord AC, | |
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APB = CPD, | 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: