Symmedian and 2 Antiparallels
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A Mathematical Droodle

7 January 2016, Created with GeoGebra


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Copyright © 1996-2018 Alexander Bogomolny

The applets illustrates the following statement:

In a triangle ABC the antiparallels to sides AB and AC that meet on the symmedian from C have equal lengths.

Symmedian and two antiparallel - problem

Let CS be the symmedian and MR and LN the two antiparallels in question that meet in point T on CS. Triangle RTN having equal base angles at R and N is isosceles. Therefore,

(1) TN = TR.

Draw the third antiparallel UV through T.

Symmedian and two antiparallel - solution

Similarly to the above, we have

TL = TV and
TM = TU.

However, as we know,

TU = TV.


MR = TM + TR
  = TL + TN
  = LN.

Note that we actually got a little more than claimed: the corresponding pieces of the equal antiparallels cut off by the symmedian are also equal.

By transitivity, the three antiparallels through the symmedian point all have equal lengths.


  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

  • What Is Antiparallel?
  • Antiparallel via Three Reflections
  • Symmedian and Antiparallel
  • Tucker Circles
  • Circle through the Incenter And Antiparallels
  • Batman's Problem
  • Antiparallel and Circumradius
  • |Activities| |Contact| |Front page| |Contents| |Geometry|

    Copyright © 1996-2018 Alexander Bogomolny


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