Optimization Problem in Acute Angle
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A Mathematical Droodle


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Explanation

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Copyright © 1996-2017 Alexander Bogomolny

This one is reminiscent of Heron's problem.

Let a point A lie in the interior of an acute angle. Find points B and C on the sides of the angle (one per side) such that the perimeter of ΔABC is minimal.

Let AB and AC be the reflections of A in the sides of the angle. Then the perimeter of ΔABC equals AB + BC + CA = ABB + BC + CAC. The latter is never shorter than the distance between AB and AC, so that

AB + BC + CA ≠ ABAC.

The equality is achieved when B and C are chosen to be the points of intersection of the corresponding sides of the angle with ABAC.

References

  1. V. M. Tikhomirov, Stories about Maximua and Minima, AMS & MAA, 1990
  2. I. M. Yaglom, Geometric Transformations I, MAA, 1962

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