Ellipse 2Given two vertices (V1 and V2) on the same axis and a line of tangency, construct the ellipse and construct the point of tangency.  For a solution to exist, the given vertices must fall on the same side of the tangent line. Construct axis V1V2, and construct the circle with diameter V1V2. Let the axis and the tangent line intersect at point A, which must be on the exterior of the circle. Let B be the inversion of A with respect to the circle. That is, AB> divides V1V2 harmonically. Construct the line through B perpendicular to V1V2, and let it intersect the tangent line at point C.  Point C is the point of tangency. The figure now has two corresponding vertices and a point on curve. Those are the given conditions for the challenge Ellipse 1, which can be used as a guide for completion of the construction. In the interactive sketch below drag the given objects to confirm that the solution holds up. |