Symmedian and the Tangents
What is this about?
A Mathematical Droodle

7 January 2016, Created with GeoGebra

Explanation

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Copyright © 1996-2018 Alexander Bogomolny

A symmedian through one of the vertices of a triangle passes through the point of intersection of the tangents to the circumcircle at the other two vertices.

symmedian passes through the intersection of two tangents

The proof is based on a known fact: the locus of the midpoints of the antiparallels to a side of a triangle is the summedian through the opposite vertex. Draw an antiparallel through S - the point of intersection of the two tangents to the circumcircle of the triangle ABC at A and B. Let it meet the extended sides AC and BC at U and V, respectively.

symmedian passes through the intersection of two tangents - solution

Then the triangles USA and VSB are isosceles, so that

SU = SA and
SV = SB.

In addition,

SA = SB,

as two tangents from a point to a circle. We conclude that S is the midpoint of UV. Therefore, S belongs to the locus of all such midpoints. Since the locus is the symmedian - a straight line - through the vertex C, CS is bound to be that symmedian.

References

  1. R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995.

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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