# Optimization Problem in Acute AngleWhat is it about? A Mathematical Droodle

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Explanation 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 ### A Sample of Optimization Problems II

• Mathematicians Like to Optimize
• Building a Bridge
• Building Bridges
• Sangaku with Quadratic Optimization
• Geometric Optimization from the Asian Pacific Mathematical Olympiad
• Cassini's Ovals and Geometric Optimization
• Heron's Problem
• Optimization in Parallelepiped
• Matrices and Determinants as Optimization Tools: an Example
• An Inequality between AM, QM and GM
• 