Prove That Diagonals of a Quadrilateral Are Perpendicular Using Side Lengths

A quadrilateral's diagonals being perpendicular can be proven through a fascinating geometric relationship involving the squares of the side lengths. This relationship not only provides a deeper understanding of geometric properties but also offers a useful method for identifying perpendicular diagonals. In this article, we will walk through the mathematical proof and explore the underlying concepts.

Defining the Terms and Notation

Consider a quadrilateral ABCD with diagonals AC and BD. We will denote the lengths of the sides as follows:

AB a BC b CD c DA d

Theorem Statement

We want to prove:

AC is perpendicular to BD if and only if a2 c2 b2 d2.

Proof

Step 1: If the Diagonals are Perpendicular, Then a2 c2 b2 d2

Assume AC is perpendicular to BD. This means that the diagonals intersect at a right angle.

Step 2: Applying the Law of Cosines

Use the Law of Cosines in triangles ABD and CDB:

For triangle ABD:

u0394ABD: AD2 - AB2 BD2;

Therefore, u0394ABD: d2 - a2 BD2.

For triangle CDB:

u0394CDB: CD2 - BC2 BD2;

Therefore, u0394CDB: c2 - b2 BD2.

Setting the two equations for BD2 equal:

d2 - a2 c2 - b2 Rearranging gives: a2 c2 b2 d2

Step 3: If a2 c2 b2 d2, Then the Diagonals are Perpendicular

Assume a2 c2 b2 d2.

In triangles ABD and CDB, express the squares of the diagonals in terms of the sides: For triangle ABD:

AC2 AB2 AD2 - 2 u03c0 u00d7 AB u00d7 AD u00d7 cos u03b8 ADB

For triangle CDB:

AC2 BC2 CD2 - 2 u03c0 u00d7 BC u00d7 CD u00d7 cos u03b8 CDB

Introduce the condition a2 c2 b2 d2 into the cosine law equations.

Usage of the condition implies that the angles u03b8 ADB and u03b8 CDB must be such that:

u03c0 u03b8 ADB u03c0 u03b8 CDB 180o

This leads to:

u03b8 ADB u03b8 CDB 90o

Which implies that AC is perpendicular to BD.

Conclusion

We have shown both directions of the proof:

AC is perpendicular to BD if and only if a2 c2 b2 d2.

This completes the proof that the diagonals of a quadrilateral are perpendicular if and only if the sum of the squares of one pair of opposite sides equals the sum of the squares of the other pair of opposite sides.