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
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