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

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.


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
Code to add this calci to your website Expand embed code Minimize embed code

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

english Calculators and Converters