Critique of reflection property

The proof of the reflection property is insightful and simple. But it assumes some facts that we haven't proved. First, how do we know that an ellipse is convex? And how do we know that the sum of distances from the foci of points outside the ellipse (whatever "outside" means) is greater than the constant sum of the ellipse? And does the Reflection Principle really mention the angles made by the path to the tangent line, or only the point on the tangent line through which the path must go?