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A cute problem on circumscribed polygons

Suppose that we have a convex polygon with some red sides and some blue sides. Suppose that it has the property that there does not exist two red sides that are adjacent, but the total lengths of the red sides combined is more than the total lengths of the blue sides.

Prove that it is impossible to inscribed within the polygon a circle.

Note by Daniel Liu
3 years, 6 months ago

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All right it has been long enough for me to post a solution.

First, suppose that we can inscribe a circle in the polygon. Note that the sum of the lengths of the blue sides is at least the sum of the red sides because of Power of a Point on each vertex. This contradicts our restriction that the length of the red sides is more than the length of the blue sides, so we are done. \(\Box\) Daniel Liu · 3 years, 6 months ago

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i think putting a diagram may help me somehow....i am lost halfway in the question....however indeed a nice question... Shubham Thakur · 3 years, 6 months ago

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Whats the prove?? Dhiraj Agarwalla · 3 years, 6 months ago

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@Dhiraj Agarwalla You will have to find that out yourself.

If it's long enough I might post one. But for now, post your ideas. Daniel Liu · 3 years, 6 months ago

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Yes it is impossible to inscribe a circle within this polygon because its impossible to make such a polygon

If u are trying to make a polygon of even number sides, number of red sticks will be equal to the number if blue sticks

If u are trying to make a polygon of odd number of sides, number of blue sticks will always be greater than number of red sticks

Its my opinion please correct me if I am wrong Shivaksh Rohatgi · 3 years, 6 months ago

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@Shivaksh Rohatgi Your argument fails for irregular polygons. Sreejato Bhattacharya · 3 years, 6 months ago

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@Guilherme Dela Corte Can you explain how that unproves it? I don't see it. Daniel Liu · 3 years, 6 months ago

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