Symmedian in a Right Triangle
What is this about?
A Mathematical Droodle

Explanation

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Copyright © 1996-2018 Alexander Bogomolny

The applet suggests a simple fact that, in a right triangle, the symmedian to the hypotenuse coincides with the altitude from the right angle.

in a right triangle, the symmedian to the hypotenuse coincides with the altitude from the right angle

A symmedian is the isogonal conjugate of a median from the same vertex. Thus, for example, in right triangle ABC, with the right angle at C, if CM is the median and CH is the symmedian through C, then angles ACM and BCH are equal. But in a right triangle, the median through the right angle equals half the hypotenuse, so that triangle AMC is isosceles. Its base angles MCA and MAC are equal. We thus have

∠CAB = ∠BCH,
∠CAB + ∠ABC = 90°, and
∠ABC = ∠HBC.

Therefore

∠HBC + ∠CBH = 90°.

It thus follows that angle CHB is right, as asserted.

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

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Copyright © 1996-2018 Alexander Bogomolny

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