Symmedian and 2 Antiparallels
What is this about?
A Mathematical Droodle

Explanation

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The applets illustrates the following statement:

Let CS be the symmedian and MR and LN the two antiparallels in question that meet in point T on CS. Triangle RTN having equal base angles at R and N is isosceles. Therefore,

Draw the third antiparallel UV through T. Similarly to the above, we have

However, as we know,

Therefore

Note that we actually got a little more than claimed: the corresponding pieces of the equal antiparallels cut off by the symmedian are also equal.

By transitivity, the three antiparallels through the symmedian point all have equal lengths.

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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