Difference between revisions of "Power of a Point Theorem"
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==== Problem 1 ==== | ==== Problem 1 ==== | ||
Two tangents from an external point <math>P</math> are drawn to a circle and intersect it at <math>A</math> and <math>B</math>. A third tangent meets the circle at <math>T</math>, and the tangents <math>\overrightarrow{PA}</math> and <math>\overrightarrow{PB}</math> at points <math>Q</math> and <math>R</math>, respectively. Find the perimeter of <math>\triangle PQR</math>. | Two tangents from an external point <math>P</math> are drawn to a circle and intersect it at <math>A</math> and <math>B</math>. A third tangent meets the circle at <math>T</math>, and the tangents <math>\overrightarrow{PA}</math> and <math>\overrightarrow{PB}</math> at points <math>Q</math> and <math>R</math>, respectively. Find the perimeter of <math>\triangle PQR</math>. | ||
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+ | ==== Problem 2 ==== | ||
+ | Square <math>ABCD</math> of side length <math>10</math> has a circle inscribed in it. Let <math>M</math> be the midpoint of <math>\overline{AB}.</math> Find the length of that portion of the segment <math>\overline{MC}</math> that lies outside of the circle. | ||
==== Other Intermediate Example Problems ==== | ==== Other Intermediate Example Problems ==== |
Revision as of 12:00, 16 June 2008
This is an AoPSWiki Word of the Week for June 13-19 |
The Power of a Point Theorem is a relationship that holds between the lengths of the line segments formed when two lines intersect a circle and each other.
Contents
Theorem
There are three possibilities as displayed in the figures below.
- The two lines are secants of the circle and intersect inside the circle (figure on the left). In this case, we have .
- One of the lines is tangent to the circle while the other is a secant (middle figure). In this case, we have .
- Both lines are secants of the circle and intersect outside of it (figure on the right). In this case, we have
Alternate Formulation
This alternate formulation is much more compact, convenient, and general.
Consider a circle and a point in the plane where is not on the circle. Now draw a line through that intersects the circle in two places. The power of a point theorem says that the product of the length from to the first point of intersection and the length from to the second point of intersection is constant for any choice of a line through that intersects the circle. This constant is called the power of point . For example, in the figure below
Notice how this definition still works if and coincide (as is the case with ). Consider also when is inside the circle. The definition still holds in this case.
Additional Notes
One important result of this theorem is that both tangents from a point outside of a circle to that circle are equal in length.
The theorem generalizes to higher dimensions, as follows.
Let be a point, and let be an -sphere. Let two arbitrary lines passing through intersect at , respectively. Then
Proof. We have already proven the theorem for a -sphere (i.e., a circle), so it only remains to prove the theorem for more dimensions. Consider the plane containing both of the lines passing through . The intersection of and must be a circle. If we consider the lines and with respect simply to that circle, then we have reduced our claim to the case of two dimensions, in which we know the theorem holds.
Problems
The problems are divided into three categories: introductory, intermediate, and olympiad.
Introductory
Problem 1
Find the value of in the following diagram:
Problem 2
Find the value of in the following diagram:
Problem 3
(ARML) In a circle, chords and intersect at . If and , find the ratio .
Problem 4
(ARML) Chords and of a given circle are perpendicular to each other and intersect at a right angle. Given that , , and , find .
Intermediate
Problem 1
Two tangents from an external point are drawn to a circle and intersect it at and . A third tangent meets the circle at , and the tangents and at points and , respectively. Find the perimeter of .
Problem 2
Square of side length has a circle inscribed in it. Let be the midpoint of Find the length of that portion of the segment that lies outside of the circle.