Cyclic Quadrilateral Ptolemy's Theorem Proof

The Theorem states that the product of the diagonals of a cyclic quadrilateral is equal to the sum of the products of opposite sides.

Cyclic Quadrilateral Ptolemy's Theorem

Given
A cyclic quadrilateral ABCD, with a,b,c,d as the length of the sides and e,f as the diagonals. Then, ac+bd = ef
The binomial coefficient (r,c)=0 if c > r.

Proof:

Given that a cyclic quadrilateral ABCD, extend CD to P such that

Since quadilateral ABCD is cyclic,

However,is also supplementary to, , so

Hence by AA similarity, and


Now, (subtend the same arc) and ,so

This yields

However,CP=CD+DP.Substituting in our expressions for CP and DP

Multiplying by AB yields

The Theorem states that the product of the diagonals of a cyclic quadrilateral is equal to the sum of the products of opposite sides.

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The online proof of Ptolemy's Theorem is made easier here.

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