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Symmedian and 2 Antiparallels: What is this about?
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In a triangle ABC the antiparallels to sides AB and AC that meet on the symmedian from C have equal lengths.
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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,
| (1) | TN = TR. |
Draw the third antiparallel UV through T. Similarly to the above, we have
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TL = TV and TM = TU. |
However, as we know,
| TU = TV. |
Therefore
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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.
The Symmedians
- Symmedian and 2 Antiparallels
- Symmedian and Antiparallel
- Symmedian in a Right Triangle
- Nobbs' Points and Gergonne Line
- Three Tangents Theorem
- A Tangent in Concurrency
- Symmedian and the Tangents
- Bride's Chair
- Star of David
- Ceva's Theorem
- The Many Ways To Construct a Triangle
- Concyclic Circumcenters: A Dynamic View
- Concyclic Circumcenters: A Sequel
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