## Pappus’s theorem

**Problem:**

Given a point on the circumference of a cyclic quadrilateral, prove that the product of the distances from the point to any pair of opposite sides or to the diagonals are equal.

**Proof:**

Let a, b, c, d be the coordinates of the vertices A, B, C, D of the quadrilateral and consider the complex plane with origin at the circumcenter of ABCD. Without loss of generality, assume that the circumradius equals 1.

The equation of line AB is

.

This is equivalent to , that is,

Let point be the foot of the perpendicular from a point M on the circumcircle to the line AB. If m is the coordinate of the point M, then

and

since .

Likewise,

,

,

and

Thus,

as claimed.

**QED.**

More later,

Nalin Pithwa

PS: The above example indicates how easy it is prove many fascinating theorems of pure plane geometry using the tools and techniques of complex numbers.

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