Simsons and 9-Point Circles in Cyclic Quadrilateral: What is this about?
A Mathematical Droodle

(When the simsons are shown, you may want to verify their definition. Click on them in turn, or click away from any to remove additional information.)

Explanation

Copyright © 1996-2008 Alexander Bogomolny

Any quadrilateral ABCD defines four triangles: BCD, CDA, DAB, ABC. In each of the triangles one may consider associate remarkable points and lines. The above applet is specifically concerned with their 9-point circles and also the Simson lines of each of the points A, B, C, D with respect to the triangle formed by the other three. The applet purports to illustrate the fact that, provided the quadrilateral ABCD is cyclic, the eight aforementioned elements all concur in a point.

First let's prove the concurrence of the four 9-point circles. As we know, the quadrilateral N_AN_BN_CN_D formed by the 9-point centers of the four triangles BCD, CDA, DAB, ABC is similar to ABCD with the factor of 1/2. So that the distance from the circumcenter of N_AN_BN_CN_D to its vertices is R/2, where R is the circumradius of ABCD. We also know that in any triangle the radius of its 9-point circle equals half that of its circumradius. It therefore follows that the circles of radii R/2 centered at the points N_A, N_B, N_C, and N_D meet at the circumcenter of the quadrilateral N_AN_BN_CN_D. In other words, the 9-point circles of the triangles BCD, CDA, DAB, ABC meet at the circumcenter of N_AN_BN_CN_D.

Now let's tackle the simsons. As we showed elsewhere, the circumcenter of N_AN_BN_CN_D coincides with H, the center of homothety that maps ABCD onto H_AH_BH_CH_D, the quadrilateral formed by the orthocenters of the triangles BCD, CDA, DAB, ABC. The two quadrilaterals ABCD and H_AH_BH_CH_D are equal so that H is the midpoint of each of the segments AH_A, BH_B, CH_C, and DH_D. By a well known property of triangles [Honsberger, p. 43], the simson of A with respect to BCD passes through the midpoint of AH_A, i.e., H. This of course applies to the other three simsons.

Note, that the result admits a natural generalization: if the quadrilateral is not cyclic, the simsons convert into circles, and the eight circles involved are still concurrent in a point.

References

1. R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995.

Copyright © 1996-2008 Alexander Bogomolny