Power of a point theorem
STILL WORKING (PLEASE DON'T EDIT YET)
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
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