Symmedian and the Tangents
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A Mathematical Droodle

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.
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|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2018 Alexander Bogomolny
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