# 1984 IMO Problems/Problem 4

## Problem

Let be a convex quadrilateral with the line being tangent to the circle on diameter . Prove that the line is tangent to the circle on diameter if and only if the lines and are parallel.

## Solution

First, we prove that if and are parallel then the claim is true: Let and intersect at (assume is closer to , the other case being analogous). Let be the midpoints of respectively. Let the length of the perpendicular from to be . It is known that the length of the perpendicular from to is . Let the foot of the perpendicular from to be , and similarly define for side . Then, since triangles and are similar, we have . This gives an expression for :

Noticing that simplifies the expression to

By the Law of Sines, . Since triangles are similar, we have and thus we have

and we are done.

Now to prove the converse. Suppose we have the quadrilateral with parallel to , and with all conditions satisfied. We shall prove that there exists no point on such that is a midpoint of a side of a quadrilateral which also satisfies the condition. Suppose there was such a . Like before, define the points for quadrilateral . Let be the length of the perpendicular from to . Then, using similar triangles, . This gives

But, we must have . Thus, we have

Since , we have as desired.