I was once looking for a construction that, given a minor axis of a conic and a tangent of this conic, finds the conic. After trying myself for some time, I asked Wilson Stothers for help, and the construction he gave was a very, very nice application of the Three-Tangents-Theorem for conics.

[TTT]Given a conic and three tangentsl,mandn, the following is true: The three lines formed by the pairwise intersections ofl,mandnand the remaining point intersect in one point.

Observing that we actually know three tangents of the conic we are looking for, we can reversely apply TTT to find additional points on the conic.

Here is the original answer of Wilson Stothers:

GIVEN Minor Axis = Segment AB, Tangent line l SUPPOSE Tangent l meets conic at D (we don't know D yet!) THEN We know center is C, the mid-point of AB We can draw tangents m at A, n at B (perpendicular to AB) Let l, m meet at X, and l, n at Y. Of course, m and n "meet" at the ideal point Z. Let BX and AY meet in G. By the Three Tangents Theorem, the line GZ (through G parallel to m and n) passes through D. AS D LIES ON l, WE HAVE FOUND D! Now draw the minor axis - the line through C perpendicular to AB If we reflect D in AB and in the minor axis, we get two further points on the conic. Call these E and F. The conic is defined by A, B, D, E and F.

And here is the construction as interactive Cinderella applet. The given elements are drawn in red, the conic is yellow. You can move A, B and l.

May 19, 2000, Ulrich Kortenkamp