home
>
theorems
>
ptolemys

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.

Related Calculators


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

 
   
  
Code to add this calci to your website Expand embed code Minimize embed code
Just copy and paste the below code to your webpage where you want to display this calculator.


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