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Help in Geometry Problem POJ-1927 Area in triangle.
By ddt, 10 years ago, In English

Can someone explain me how to solve this problem http://poj.org/problem?id=1927 If the length of rope is smaller than the perimeter of circle with radius equal to radius of inscribed circle, we can directly find the answer. But, how to deal with cases when length of the rope is greater than perimeter of such circle and less than perimeter of triangle.

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10 years ago, # |
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I don't know how to solve this problem. I try to guess.
The shape of the resulting figure looks like an triangle with cutted anlges (by a line perpendicular to it's bisector) and with three circular segments attached to cutted angles. circular segment's radiuses are equal to radius of inscribed circle. Imagine you are inflating the balloon. I suppose after step where perimeter of ballon is equal to perimeter of inscribed circle biggest segment will move towards it's angle. Binary search length of the biggest chord.

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    10 years ago, # ^ |
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    I think I was wrong. It seems to me described segments always forms an circle, so the triangles are similar. Second triangle I mentioned is formed from cutted angles.

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      10 years ago, # ^ |
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      I think the cutted angle-segments must be part of a circle which should be incircle for a triangle similar to the given triangle and having larger sides. But, I am unable to derive a relation or equation relating these two triangles.

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        10 years ago, # ^ |
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        length of the rope is a — a' + b — b' + c — c' + 2*Pi*R'
        R' — radius of a circle inscribed in a smaller similar triangle.

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