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Proof Contest Day 2 (Follow Up problem)

The previous problem dealt with proving that the sum of heights remains constant. However , in this follow up problem we would find the exact value of the sum of heights for an even sided polygon.


PROBLEM:

Let \(P\) be any point in the interior of a regular polygon of \(2n\) sides. Perpendiculars \(PA_1 \ , \ PA_2 \ , \ PA_3 \ , \ \dots \ , \ PA_{2n}\) are drawn to the sides of the polygon. Show that: \(\displaystyle\sum_{i=1}^{2n} PA_{i} = 2nr\) where \(r\) is the radius of inscribed circle of polygon.

Also show that \(\displaystyle\sum_{i=1}^{n} PA_{2i-1} =\sum_{j=1}^{n} PA_{2j} =nr\)


This problem is not original

Note by Nihar Mahajan
11 months, 1 week ago

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The first one can be pretty easily proved using the fact that the radius of the inscribed circle is the height of the triangle.The second one can be proved using the fact that since the polygpn has an even number of sides the two perpendiculars,\(PA_{i},PA_{i+n}\) form a straight line,from here we easily get that,\[PA_{i}+PA_{i+n}=PA_{k}+PA_{k+n}\],now just put values,\(1,3,5...(n-1)\) for i and the rest for k and then add to get the result,i am writing this on phone so cant explain very clearly. Adarsh Kumar · 11 months, 1 week ago

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@Adarsh Kumar Exactly! Nice use of the fact that since the polygon has an even number of sides the two perpendiculars, \(PA_i , PA_{i+n}\) form a straight line. Nihar Mahajan · 11 months, 1 week ago

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@Nihar Mahajan Thanx!Sorry for the late reply. Adarsh Kumar · 11 months, 1 week ago

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We know that the sum of the distances of the perpendiculars from the interior point is always constant. (As proved earlier). So we know that \[PA_1+...+PA_{2n}=k\] a constant.

Now the incentre of the polygon is also such a point so the sum of the perpendiculars from it will be the sum of \(2n\) radii. So we get that \[PA_1+PA_2+...+PA_{2n}=2nr=k\] Aditya Agarwal · 11 months, 1 week ago

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@Nihar Mahajan What do you mean? Aditya Agarwal · 11 months, 1 week ago

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@Aditya Agarwal Oh wait , I got your point. Can you think of some other way to prove it? Nihar Mahajan · 11 months, 1 week ago

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@Nihar Mahajan Is there an even simpler one? I believe my method wasn't the solution you had in mind? Aditya Agarwal · 11 months, 1 week ago

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@Aditya Agarwal I had solved this question about an year ago.I don't have solution ready , but I know the basic theme Nihar Mahajan · 11 months, 1 week ago

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@Aditya Agarwal Try proving \(\displaystyle\sum_{i=1}^{n} PA_{2i-1} =\sum_{j=1}^{n} PA_{2j} =nr\) . Nihar Mahajan · 11 months, 1 week ago

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@Nihar Mahajan See the edit. You misunderstood my solution. Aditya Agarwal · 11 months, 1 week ago

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Can we directly prove the last thing? Aditya Agarwal · 11 months, 1 week ago

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@Aditya Agarwal Yes. Then the main problem becomes obvious from there. Nihar Mahajan · 11 months, 1 week ago

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