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

Explanation

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. Then the triangles USA and VSB are isosceles, so that

In addition,

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.

The Symmedians

1. Symmedian and 2 Antiparallels
2. Symmedian and Antiparallel
3. Symmedian in a Right Triangle
4. Nobbs' Points and Gergonne Line
5. Three Tangents Theorem
6. A Tangent in Concurrency
7. Symmedian and the Tangents
8. Bride's Chair
9. Star of David
10. Ceva's Theorem
11. The Many Ways To Construct a Triangle
12. Concyclic Circumcenters: A Dynamic View
13. Concyclic Circumcenters: A Sequel