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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, 5 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 $$\sqrt{\sqrt{3} xy + x^2 + y^2}$$. · 1 year, 5 months ago

Can you post the method you used to find the solution? · 1 year, 5 months ago

I have updated the Wiki · 1 year, 5 months ago

Great $$:)$$ · 1 year, 5 months ago