The Taylor Circle

In 1976, Ross Honsberger proposed a problem (#192) in Eureka, a predecessor of Crux Mathematicorum of today (I am grateful to W.McWorter who copied the Eurika's page for me.)

As the editor of Eureka commented, the circle was little known at the time (1976). It is still today, as I found it only mentioned once (and that in passing) in a discussion on Conway's circle.

The circle is named after H.M.Taylor (1842-1927) who discussed it in 1882. The editor of Eureka remarked further that

Let H be the orthocenter of ABC. The diagram exhibits a great deal of similar triangles. We start with the following three:

From the similarities, we derive

Thus

and

The latter is the converse of Proposition III.35 from Euclid's Elements, namely

(The rectangle in the statement is in fact a rectangle's area, or the product of two segments. This product is the power of the point of intersection with respect to the circle.)

Euclid did not mention this, but the converse to Proposition III.35 is also true:

Assume, on the contrary, that the circle that passes through D, F, and G does not contain E. It then crosses line DE at some point E', for which, as Euclid tells us, PD·PE' = PF·PG. But then PE = PF·PG/PD = PE' so that E = E'. Contradiction.

Back to our original problem, (*) implies that the four points H_ba, H_ca, H_ab, and H_cb lie on a circle. Call it Circle_c. Similarly, the four points H_ab, H_cb, H_bc, and H_ac lie on a circle, say, Circle_a, while points H_ba, H_ca, H_bc, and H_ac lie on a circle Circle_b.

We would like to show that all three circles coincide. This will be true if any two of them coincide. Thus, let's assume that all three are different. Note that then AB serves as the radical axes of Circle_a and Circle_b, and the same is true with regard the sides BC and AC and circles Circle_b and Circle_c and Circle_a and Circle_c, respectively. As we know, the radical axes of three circles meet at a point, which is absurd unless ABC shrinks to a point. Therefore, we conclude that the three circles Circle_a, Circle_b, and Circle_c indeed coincide, so that all six points, H_ab , H_ba , H_ca , H_ac , H_bc , H_cb, do lie on a circle.

Copyright © 1996-2008 Alexander Bogomolny