Ptolemy's Theorem
Ptolemy's Theorem gives a relationship between the side lengths and the diagonals of a cyclic quadrilateral; it is the equality case of the Ptolemy Inequality. Ptolemy's Theorem frequently shows up as an intermediate step in problems involving inscribed figures.
Contents
Definition
Given a cyclic quadrilateral with side lengths
and diagonals
:
.
Proof
Given cyclic quadrilateral extend
to
such that
Since quadrilateral is cyclic,
However,
is also supplementary to
so
. Hence,
by AA similarity and
Now, note that (subtend the same arc) and
so
This yields
However, Substituting in our expressions for
and
Multiplying by
yields
.
Problems
Equilateral Triangle Identity
Let be an equilateral triangle. Let
be a point on minor arc
of its circumcircle. Prove that
.
Solution: Draw ,
,
. By Ptolemy's Theorem applied to quadrilateral
, we know that
. Since
, we divide both sides of the last equation by
to get the result:
.
Regular Heptagon Identity
In a regular heptagon ABCDEFG, prove that: 1/AB = 1/AC + 1/AD.
Solution: Let ABCDEFG be the regular heptagon. Consider the quadrilateral ABCE. If a, b, and c represent the lengths of the side, the short diagonal, and the long diagonal respectively, then the lengths of the sides of ABCE are a, a, b and c; the diagonals of ABCE are b and c, respectively.
Now, Ptolemy's Theorem states that ab + ac = bc, which is equivalent to 1/a=1/b+1/c.
1991 AIME Problems/Problem 14
A hexagon is inscribed in a circle. Five of the sides have length and the sixth, denoted by
, has length
. Find the sum of the lengths of the three diagonals that can be drawn from
.