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Here's a question that I found on the website of International Kangaroo Maths Contest. The question goes like this:

What is the total number of right angled triangles that can be formed by joining the vertices of a regular 14-gon?

Correct Answer: The correct answer as given on the website is $84$.

What I did:

First of all, I calculated the total number off triangles (right angled or not) that can be formed by using the vertices of a regular 14-gon that is $^{14}C_3=364$.

However, I can't figure out how to count which ones of these $364$ triangles are right angled. Please help me in this regard.

Thanks for the attention!

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1 Answer

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Remember that if two points of triangle are makeing diameter of circle circumscribed to this triangle than this one is right.

We have $7$ diameters and for each one we have 6 point on each side. So we have $$7\cdot 12=84$$ right triangles.

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