Problem - 03 - National Math Camp 2021 Combinatorics Test - "All you see is isosceles"

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Anindya Biswas
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Problem - 03 - National Math Camp 2021 Combinatorics Test - "All you see is isosceles"

Unread post by Anindya Biswas » Fri Apr 30, 2021 5:25 pm

Show that for all positive integers $n\geq8$, you can cut any quadrilateral into $n$ isosceles triangles.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

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Mehrab4226
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Re: Problem - 03 - National Math Camp 2021 Combinatorics Test - "All you see is isosceles"

Unread post by Mehrab4226 » Sat May 01, 2021 5:19 pm

First, let me give a claim,

Claim-1:
For all $n\geq 4$ we can cut a triangle into $n$ isosceles triangles.
Proof:
Base Case:
Let us take the largest angle of a given triangle and drop an altitude on the opposite side. The altitude must be inside the triangle. So our initial triangle has become $2$ right-angled triangles. Which we can cut into $4$ isosceles triangles by cutting the line from the circumcentre of each right-angled triangle and its vertices. So our base case is done.
Inductive hypothesis:
Let we can cut any given triangle in $k$ isosceles triangle.
Inductive step:
Let, $\triangle ABC$ be a triangle and where $AC$ is its longest side. Now we take $A$ or $C$ as the centre and $AB$(If we took $A$) or $BC$(If we took $B$) as radius and draw a circle that intersects $AC$ at $X$. Now we can cut through $BC$ and get an isosceles triangle $AXB$ if we chose $A$ or $CBX$ if we chose $C$. Depending on our choice one of them is an isosceles triangle and the other is another triangle which we can cut into $k$ isosceles triangle[By our inductive hypothesis]. So we can cut $\triangle ABC$ into $k+1$ isosceles triangles. $\square$

Now the rest is easy. We just draw a diagonal of the quadrilateral and have 2 triangles which we can cut into any number of isosceles triangles $k\geq 4$ by claim-1. So we can cut any quadrilateral in $n \geq 8$ triangles. $\square$
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

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