Outline Mathematics
Geometry

Dan Pedoe's Observation

The Urquhart's Theorem"most elementary theorem" of Euclidean geometry has been discovered and so named by the Australian mathematician M. L. Urquhart (1902-1966). It was popularized by Dan Pedoe who found in 1976 an equivalent formulation:

Pedoe further wrote

Following the applet, I'll give Pedoe's proof based on a nice lemma by Basil Rennie. Next, I'll show an additional feature of the configuration which Pedoe may have overlooked.

To prove the theorem, let S_C be a circle through E and F tangent to CD at U, and suppose that S_A touches AD at and AB at .

We wish to show that S_C is tangent to BC.

Since both cirlces path through E and F, EF is their . Thus B and D lie on the radical axis of the two circles. Since D lies on the radical axis of the two circles,

We proceed as follows

To sum up, we see that

or

Now recollect that B also lies on the radical axis of S_A and S_C so that BH equals the tangent from B to S_C. Thus (2) can be read as the distance between two points (B and C) being equal to the difference in the length of the tangents from each to a circle (S_C). This is exactly the context of

With the help of the lemma, (2) implies that BC is tangent to S_C. This proves Pedoe's Theorem.

Let V be the point of tangency of BC and S_C. We can observe that the four points U, G, H, V are collinear.

Let's denote the internal parallelogram angles at A and C as a. As we know, triangles UDG, GAH, and HBV are all with the angle at the vertex equal to a. Their base angles are all equal to 90^o - a/2. Therefore, at G and H, we have pairs of equal and thus vertical angles. Which exactly means that GH is a continuation of EG and of HV.

References

1. J. Konhauser, D. Velleman, S. Wagon, Which Way Did the Bicycle Go?, MAA, 1996, #73
2. D. Pedoe, The Most "Elementary" Theorem Of Euclidean Geometry, Math Magazine, vol 49, no 1 (Jan., 1976), 40-42