Department of Mechanical Engineering, University of Applied Sciences, Dortmund, Germany.
Keywords: Ptolemy's theorem; vectorial proof; cyclic quadrilateral; geometry;
In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle) .
With given side and diagonal lengths, Ptolemy's theorem of a cyclic quadrilateral states:
For a cyclic quadrilateral the product of the diagonal lengths is equal to the sum of the product of the length of the pairs of opposite sides.
The quadrilateral can be described by a loop closure of side vectors , , , running counter-clockwise (Fig.1) .
Expressing the diagonal vector by the sides of its two triangles and squaring leads us to
Both angles in the triangle corners opposite to diagonal sum up to 180° and thus equals . So we can write in vectorial notation 
Resolving for the dot products and in equation (2) and introducing them in (3) gives
On our way for isolating we get
Herein the right hand side can be simplified by resolving the brackets and reordering
Otherwise expressing the diagonal vector by its two triangle sides and squaring again gives
Both angles in the triangle corners opposite to diagonal also sum up to 180° and equal to . Then using this fact in an analog manner leads us to
Now multiplying equations (4) and (5), reducing the fractions and applying the square root finally results in (1)