As a point traces the ellipse, the line segments joining the point to the circles are equal in length to the line segments joining the point to the respective foci. The moving point is also joined to the directrices. Next, to see that the ratio of the distances to the foci and the directrices is constant, the lines are projected vertically up and down (as applicable) to the planes of the circles of contact to form triangles. Finally, since the triangles do not change shape as they move, the ratios must be constant, and the two definitions (cone cut by a plane and constant distance to focus-distance to directrix ratio) are seen to be equivalent. Moreover, the third definition (constant sum of distances to the foci) can also be seen.