Symmedian and Antiparallel
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A Mathematical Droodle

2 September 2015, Created with GeoGebra

Explanation

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Copyright © 1996-2018 Alexander Bogomolny

The applet illustrates the following fact:

A symmedian drawn from a vertex of a triangle divides the antiparallels to the opposite side in half.

a property of symmedians

Indeed, let CS be the symmedian from vertex C and UV an antiparallel to the side AB. By the definition of the antiparallel, the triangles ABC and VUC are similar. Assume that this similarity maps CM to CT. Then the segments UT and VT are equal as are the angles ACM and VCT. It follows that CT is a part of the reflection of CM in the angle bisector of C. But this is exactly the definition of the symmedian. It follows that CS (CT extended) is the symmedian from C.

This result is often presented in a different form: the locus of the midpoints of the antiparallels to a side of a triangle is the summedian through the opposite vertex.

Corollary

In a right-angled triangle the symmedian point coincides with the midpoint of the altitude to the hypotenuse. Indeed, let in ΔABC angle B be right. Then the altitude BHB is antiparellel to both AB and BC. Therefore, it is halved by either of them. In other words, the midpoint of BHb lies on the symmedians through A and C. Since the symmedian point is the point of concurrency of the three symmedians, it is also the point where any two of them meet.

Symmedian

  1. All about Symmedians
  2. Symmedian and Antiparallel
  3. Symmedian and 2 Antiparallels
  4. Symmedian in a Right Triangle
  5. Nobbs' Points and Gergonne Line
  6. Three Tangents Theorem
  7. A Tangent in Concurrency
  8. Symmedian and the Tangents
  9. Ceva's Theorem
  10. Bride's Chair
  11. Star of David
  12. Concyclic Circumcenters: A Dynamic View
  13. Concyclic Circumcenters: A Sequel
  14. Steiner's Ratio Theorem
  15. Symmedian via Squares and a Circle
  16. Symmedian via Parallel Transversal and Two Circles
  17. Symmedian and the Simson
  18. Characterization of the Symmedian Point with Medians and Orthic Triangle
  19. A Special Triangle with a Line Through the Lemoine Point

Related material
Read more...

  • What Is Antiparallel?
  • Antiparallel via Three Reflections
  • Tucker Circles
  • Symmedian and 2 Antiparallels
  • Circle through the Incenter And Antiparallels
  • Batman's Problem
  • Antiparallel and Circumradius
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    Copyright © 1996-2018 Alexander Bogomolny

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