Power of a point theorem

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Theorem:

There are three unique cases for this theorem. Each case expresses the relationship between the length of line segments that pass through a common point and touch a circle in at least one point.

Case 1 (Inside the Circle):

If two chords $AB$ and $CD$ intersect at a point $P$ within a circle, then $AP\cdot BP=CP\cdot DP$

Case 2 (Outside the Circle):

Classic Configuration
Tangent Line

Normal Configuration

Tangent Line

Case 3 (On the Border/Useless Case):

    • Still working

Proof

Problems

Introductory (AMC 10, 12)

Intermediate (AIME)

Olympiad (USAJMO, USAMO, IMO)