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.

28 November 2015, Created with GeoGebra

Explanation

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Copyright © 1996-2017 Alexander Bogomolny

The six feet of the perpendiculars to the sides of triangle ABC dropped from a point P and its isogonal conjugate Q are concyclic, i.e. all six lie on the same circle.

28 November 2015, Created with GeoGebra

Let the feet of the perpendiculars from P to BC, CA and AB be AP, BP, CP, and similarly define AQ, BQ, CQ. The above states that the circumcircles of triangles APBPCP and AQBQCQ coincide. In other words, we arrive at the following

Theorem

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

Proof

The quadrilateral PBPACP is cyclic, for two opposite angle at BP and CP both measuring 90° add up to 180°. Therefore two angles PBPCP and PACP that subtend the same chord PCP are equal. Similarly, ∠QABQ = ∠QCQBQ. Since P and Q are isogonal conjugate, we also have ∠PACP = ∠QABQ. Therefore, ∠PBPCP = ∠QCQBQ. Subtracting these from the equal angles PBPA and QBCA we obtain ∠CPBPBQ = ∠BQCQCP. The latter angles subtend the same segment BQCP, which proves that the four points BP, BQ, CP, CQ are concyclic - they lie on the same circle. The center of the circle is necessarily the point of intersection of the perpendicular bisectors to BPBQ and CPCQ, which is the midpoint of the segment PQ.

The above argument is completely symmetric with respect to the sides of ΔABC. So it follows that all six points AP, AQ, BP, BQ, CP and CQ lie on a circle centered at the midpoint of PQ.

As we know, if P lies on the circumcircle of ΔABC, 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 ΔABC, 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 ΔABC.

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

Related material
Read more...

Simson Line - the simson

  • Simson Line: Introduction
  • Simson Line
  • Three Concurrent Circles
  • 9-point Circle as a locus of concurrency
  • Miquel's Point
  • Circumcircle of Three Parabola Tangents
  • Angle Bisector in Parallelogram
  • Simsons and 9-Point Circles in Cyclic Quadrilateral
  • Reflections of a Point on the Circumcircle
  • Simsons of Diametrically Opposite Points
  • Pentagon in a Semicircle
  • Simson Line in Disguise
  • Two Simsons in a Triangle
  • Carnot's Theorem
  • A Generalization of Simson Line

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