Symmedian and Antiparallel
What is this about?
A Mathematical Droodle

Explanation

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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.

Corollary

In a right-angled triangle the symmedian point coincides with the midpoint of the altitude to the hypotenuse. Indeed, let in ΔABC angle B be right. Then the altitude BH_B is antiparellel to both AB and BC. Therefore, it is halved by either of them. In other words, the midpoint of BH_b lies on the symmedians through A and C. Since the symmedian point is the point of concurrency of the three symmedians, it is also the point where any two of them meet.

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

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