Proof of reflection property of ellipse

This proof is due to Zalman P. Usiskin. Since an ellipse is convex, all points on any tangent line, except the point of tangency, lie outside the ellipse. Therefore the sum of distances from any point on the tangent line, other than the point of tangency, is greater than the constant sum for the ellipse. In other words, the shortest path from one focus to the other by way of the tangent line is through the point of tangency. By the Reflection Principle, the angles made between the tangent line and the segments of that path are equal.