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A Random Point

Let \(\Delta ABC\) be an equilateral triangle and \(P\) be a point inside this triangle such that \(PA=x,PB=y\) and \(PC=z\), If \(z^2=x^2+y^2\), find the length of the sides of \(\Delta ABC\) in terms of \(x\) and \(y\).

This is from a local Olympiad entrance exam.

Note by Sualeh Asif
1 year, 8 months ago

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General formula can be worked out to be \[\sqrt{(4 \sqrt{3} A + a^2 + b^2 + c^2)/2}\] where a, b, c are PA, PB and PC and A is a area of a triangle with side lengths a, b and c. In this case x, y, z form right angled triangle and that means that side length of the equilateral triangle is \(\sqrt{\sqrt{3} xy + x^2 + y^2}\). Maria Kozlowska · 1 year, 8 months ago

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@Maria Kozlowska Can you post the method you used to find the solution? Sualeh Asif · 1 year, 7 months ago

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@Sualeh Asif I have updated the Wiki Maria Kozlowska · 1 year, 7 months ago

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@Maria Kozlowska Great \(:)\) Sualeh Asif · 1 year, 7 months ago

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Where did you got it from? Please name it. Akshat Sharda · 1 year, 8 months ago

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