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 $AP\cdot BP=CP\cdot DP$

$ $[asy] draw(circle((0,0),5)); [/asy]$ $===Case 2 (Outside the Circle):===

=====Classic Configuration=====

Given lines$ (Error compiling LaTeX. Unknown error_msg) 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$ (Error compiling LaTeX. Unknown error_msg) AB $and$ AC $with$ AC $[[tangent line|tangent]] to the related circle at$ 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,$ (Error compiling LaTeX. Unknown error_msg) 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):

Proof

Problems

Introductory (AMC 10, 12)

Intermediate (AIME)

Olympiad (USAJMO, USAMO, IMO)