Nuruhuhuhu's blog

By Nuruhuhuhu, history, 16 months ago, In English,

Let ABC be a non-degenerate triangle, let P be the point such that, minimum of angles APB, BPC, CPA is maximized. Is P a special point? Is it guaranteed to be inside the triangle?

 
 
 
 
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16 months ago, # |
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    16 months ago, # ^ |
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    Thank you. I knew about Fermat point though. But one of the angles of the triangle exceeds 120, does it work then too? The main idea of Fermat point is to minimize the sum of the distances from the vertices to the point. When an angle exceeds 120, Fermat point becomes one of the vertices. But then the minimum angle becomes 0. We can construct the Fermat point outside the triangle when the angle exceeds 120. It violates the sum of distances to be minimum property, but does it satisfy the angle property?

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      16 months ago, # ^ |
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      If one of the angles a > 2·π / 3 then the best we can get is π - a / 2 - ε for any ε > 0. It will be almost in this vertex and chosen such that angles to closest sides are equal. It is always possible by continuity.