Simson Line in Disguise

The figure below shows a triangle ABC with a point D. DE is perpendicular to DC (E is on BA extended), DF is perpendicular to DA (F is on BC extended), and DG is perpendicular to DB (G is on AC extended). Prove that points E, F, G are collinear.


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Solution

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Copyright © 1996-2017 Alexander Bogomolny

The figure below shows a triangle ABC with a point D. DE is perpendicular to DC (E is on BA extended), DF is perpendicular to DA (F is on BC extended), and DG is perpendicular to DB (G is on AC extended). Prove that points E, F, G are collinear.


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Solution

Let C(Δ) denote the circumcircle of triangle Δ. Draw C(DEF), C(DFG), and C(DGE). Let their centers be C', A', and B', respectively.

Consider a pair of them, say C(DFG), and C(DGE). The two circles meet at two points D and G. Therefore, their center line B'C' is perpendicular to the common chord DG and passes through the midpoint Gm of the latter. Similar considerations apply to the two remaining pairs of the circles.

At this point, I wish to strengthen the statement. From the forgoing observations, the sides of ΔA'B'C' are parallel to the cevians AD, BD, and CD. Which brings to mind the Maxwell theorem. According to that theorem, the cevians in ΔA'B'C' parallel to the sides of ΔABC are concurrent. Let's denote the point of concurrency D'. Then we have

The four points A, F, G, and D' are concurrent.

... to be continued ...

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
  • Simson Line From Isogonal Perspective
  • Pentagon in a Semicircle
  • Two Simsons in a Triangle
  • Carnot's Theorem
  • A Generalization of Simson Line

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