Urquhart's Theorem For And By Conics

Urquhart's theorem states that, for straight lines ABB' and AC'C with BC and B'C' intersecting in D the relation AB + BD = AC' + C'D implies AB' + B'D = AC + CD, as when four lines displayed all touch the same circle.

Michel Cabart wondered whether an "hyperbolic" version of the theorem is also true:

What follows is the combination of Michel's emails of December 9 and 15, 2008.

Using a trigonometric approach in analogy to the "circular theorem":

by posing t = tanβ' / tanβ. Thus BD - BA = C'D - C'A iff tanβ' / tanβ = tanγ / tanγ' and the rest of the proof is similar.

The two variants of the theorem can be summarized into one single property: "If B and C' belong to an ellipse (resp. hyperbola), then B' and C also belong an ellipse (resp. hyperbola) with the same foci."

This suggests a proof by conics theorems. The following one (detailed for ellipse and similar for hyperbola) uses the following property of the focuses: for any two tangents MA and CA intersecting in A, FA bisects MFC and F'A bisects MF'C. Here we take CA perpendicular to the major axis.

Notations:

• ∠ MF'C = 2α' and ∠MFC = 2α,
• c = half the distance between foci, a = half the length of the major axis.

We then have:

so that

Thus returning to the problem: BD + BA = C'D + C'A iff

which is the condition found by trigonometric calculations.

Urquhart's Theorem

• Urquhart's Theorem
• Proof by M. Cabart
• An elementary synthetic solution
• From Leningrad Mathematical Olympiad
• Pedoe's theorem
• Pitot theorem