Simson Line From Isogonal Perspective

If a point is selected on the circumcircle of a triangle and the perpendiculars are dropped from that point to the sides of the triangle, the feet of the perpendiculars are colinear. The line to which the three belong is known as the Simson line (of the selected point with respect to the triangle.)

The applet below illustrates a property of isogonal conjugates that generalizes the Simson line.

Explanation

Copyright © 1996-2009 Alexander Bogomolny

Let the feet of the perpendiculars from P to BC, CA and AB be A_P, B_P, C_P, and similarly define A_Q, B_Q, C_Q. The above states that the circumcircles of triangles A_PB_PC_P and A_QB_QC_Q coincide. In other words, we arrive at the following

Theorem

The pedal circles of a point and its isogonal conjugate coincide.

Proof

The quadrilateral PB_PAC_P is cyclic, for two opposite angle at B_P and C_P both measuring 90^o add up to 180^o. Therefore two angles PB_PC_P and PAC_P that subtend the same chord PC_P are equal. Similarly, QAB_Q = QC_QB_Q. Since P and Q are isogonal conjugate, we also have PAC_P = QAB_Q. Therefore, PB_PC_P = QC_QB_Q. Subtracting these from the equal angles PB_PA and QB_CA we obtain C_PB_PB_Q = B_QC_QC_P. The latter angles subtend the same segment B_QC_P, which proves that the four points B_P, B_Q, C_P, C_Q are concyclic - they lie on the same circle. The center of the circle is necessarily the point of intersection of the perpendicular bisectors to B_PB_Q and C_PC_Q, which is the midpoint of the segment PQ.

The above argument is completely symmetric with respect to the sides of ABC. So it follows that all six points A_P, A_Q, B_P, B_Q, C_P and C_Q lie on a circle centered at the midpoint of PQ.

As we know, if P lies on the circumcircle of ABC, Q is a point at infinity. In that case, the radius of the above circle is infinite, and the circle appears to be a line. Such that, if P lies on the circumcircle of ABC, the feet of the three perpendiculars from P to the sides of the triangle lie on that straight line. The line is known as the Simson line of P with respect to ABC.

References

1. R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995, pp. 67-68
2. V. V. Prasolov, Essays On Numbers And Figures, AMS, 2000, pp. 65-69

Copyright © 1996-2009 Alexander Bogomolny