Let $M$ be the midpoint of side $AB$ of triangle $ABC$. Prove that $CM=AB/2$ if and only if angle $\angle ACB = 90^\circ$.
Let $M$ be the midpoint of side $AB$ of triangle $ABC$. Prove that $CM=AB/2$ if and only if angle $\angle ACB = 90^\circ$.
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geometry
1 Answers
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Hint: By definition of $M$ the condition is equivalent to $MA=MB=MC$ i.e. $M$ being the circumcenter of $\triangle ABC$. On the other hand, $\angle ACB = 90^\circ$ is equivalent to $BC$ being a diameter of the circumcircle.
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0Has it something to do with triangle's area? – 2017-01-22
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0@Estonian No, simply with the [inscribed angle](https://en.wikipedia.org/wiki/Inscribed_angle). – 2017-01-22
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1Solved it! :) Thanks for hint, – 2017-01-22