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Isosceles triangle and the length of parallel segments

There is a isosceles triangle ABC. Point P is on the base BC.
Two lines which are parallel to AB and AC respectively and intersect on P are drawn.
The intersections of parallel lines with AB and AC are point Q and R respectively.
Then PQ + PR is constant.

  1. : Origical theorem
  2. : If not segment BC but line BC
  3. : If not isosceles triangle

Forum of Geometric Constructor by Y.Iijima