Power of a point theorem

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 $AP\cdot BP=CP\cdot DP$

$[asy] draw(circle((0,0),3)); dot((0,0)); dot((-2.93, 1.1)); label("A",(-2.82,1)); [/asy]$

Case 2 (Outside the Circle):

Classic Configuration

Given lines $AB$ and $CB$ originate from two unique points on the circumference of a circle ( $A$ and $C$ ), intersect each other at point $B$ , outside the circle, and re-intersect the circle at points $F$ and $G$ respectively, then $BF\cdot BA=BG\cdot BC$

Tangent Line

Given Lines $AB$ and $AC$ with $AC$ tangent to the related circle at $C$ , $A$ lies outside the circle, and Line $AB$ intersects the circle between $A$ and $B$ at $D$ , $AD\cdot AB=AC^{2}$

Case 3 (On the Border/Useless Case):

If two chords, $AB$ and $AC$ , have A on the border of the circle, then the same property such that if two lines that intersect and touch a circle, then the product of each of the lines segments is the same. However since the intersection points lies on the border of the circle, one segment of each line is $0$ so no matter what, the constant product is $0$ .

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Power of a point theorem

Contents

Theorem:

Case 1 (Inside the Circle):

Case 2 (Outside the Circle):

Classic Configuration

Tangent Line

Case 3 (On the Border/Useless Case):

Proof

Problems

Introductory (AMC 10, 12)

Intermediate (AIME)

Olympiad (USAJMO, USAMO, IMO)