Difference between revisions of "Power of a Point Theorem"
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==== Problem 3 ==== | ==== Problem 3 ==== | ||
− | ([[ARML]]) In a circle, chords <math>AB</math> and <math>CD</math> intersect at <math>R</math>. If <math>AR:BR=1:4</math> and <math>CR:DR=4:9</math>, find the ratio <math>AB:CD</math>. | + | ([[ARML]]) In a circle, chords <math>AB</math> and <math>CD</math> intersect at <math>R</math>. If <math>AR:BR=1:4</math> and <math>CR:DR=4:9</math>, find the ratio <math>AB:CD</math> . |
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==== Problem 2 ==== | ==== Problem 2 ==== | ||
− | Square <math>ABCD</math> of side length <math>10</math> has a circle inscribed in it. Let <math>M</math> | + | 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 ==== |
Latest revision as of 14:30, 30 June 2024
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 chords 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
Hint for Proof
Draw extra lines to create similar triangles! (Hint: Draw on all three figures. Draw another line as well.)
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 at point
. 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 (this means that T is on the minor arc
). If
, 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.