Symmedian and Antiparallel
What is this about?
A Mathematical Droodle

Explanation

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The applet illustrates the following fact:

Indeed, let CS be the symmedian from vertex C and UV an antiparallel to the side AB. By the definition of the antiparallel, the triangles ABC and VUC are similar. Assume that this similarity maps CM to CT. Then the segments UT and VT are equal as are the angles ACM and VCT. It follows that CT is a part of the reflection of CM in the angle bisector of C. But this is exactly the definition of the symmedian. It follows that CS (CT extended) is the symmedian from C.

This result is often presented in a different form: the locus of the midpoints of the antiparallels to a side of a triangle is the summedian through the opposite vertex.

Corollary

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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