# Optimization Problem in Acute Angle

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A Mathematical Droodle

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Copyright © 1996-2018 Alexander BogomolnyThis 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 A_{B} and A_{C} be the reflections of A in the sides of the angle. Then the perimeter of ΔABC equals _{B}B + BC + CA_{C}._{B} and A_{C}, so that

AB + BC + CA ≠ A_{B}A_{C}.

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

### References

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

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Copyright © 1996-2018 Alexander Bogomolny