# Power of a point theorem

## Contents

## 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. Can be useful with cyclic quadrilaterals as well however with a slightly different application.

### Case 1 (Inside the Circle):

If two chords and intersect at a point within a circle, then

### Case 2 (Outside the Circle):

##### Classic Configuration

Given lines and originate from two unique points on the circumference of a circle ( and ), intersect each other at point , outside the circle, and re-intersect the circle at points and respectively, then

##### Tangent Line

Given Lines and with tangent to the related circle at , lies outside the circle, and Line intersects the circle between and at ,

### Case 3 (On the Border/Useless Case):

If two chords, and , have 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 so no matter what, the constant product is .

## Proof

### Case 1 (Inside the Circle)

Join and .

In

(Angles subtended by the same segment are equal)

(Vertically opposite angles)

(Corresponding sides of similar triangles are in the same ratio)

### Case 2 (Outside the Circle)

Join and

(Why?)

Now, In

(shown above)

(common angle)

(Corresponding sides of similar triangles are in the same ratio)

### Case 3 (On the Circle Border)

Length of a point is zero so no proof needed :)

## Problems

#### Introductory (AMC 10, 12)

Let be a diameter in a circle of radius Let be a chord in the circle that intersects at a point such that and What is

Source: 2020 AMC 12B Problems/Problem 12

#### Intermediate (AIME)

Let be a triangle inscribed in circle . Let the tangents to at and intersect at point , and let intersect at . If , , and , can be written as the form , where and are relatively prime integers. Find .

Source: 2024 AIME I Problems/Problem 10

#### Olympiad (USAJMO, USAMO, IMO)

Given circles and intersecting at points and , let be a line through the center of intersecting at points and and let be a line through the center of intersecting at points and . Prove that if and lie on a circle then the center of this circle lies on line .

Source: 2009 USAMO Problems/Problem 1

Let be a point interior to triangle (with ). The lines , and meet again its circumcircle at , , respectively . The tangent line at to meets the line at . Show that from follows .

Source: 2010 IMO Problems/Problem 4