In the figure is a triangle with an inscribed circle. Plot a parallel line to side a, which passes through the center of the circle.

Since the segment lines OM and ON are the hypotenuses of the right triangle formed with radius of the circle.

OM> r

ON >r

Addition of the two inequalities gives

OM+ON>2r

Since the sum of OM and ON is the segment MN, which is always less than side a in length.

Now we have proved that the diameter of the inscribed circle of a triangle is less than the length of any side of the triangle.