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.

#Geometry #Triangles #Points #Olympiad #MadMath

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

Where did you got it from? Please name it.

Akshat Sharda - 5 years, 10 months ago

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 - 5 years, 10 months ago

Can you post the method you used to find the solution?

Sualeh Asif - 5 years, 10 months ago

I have updated the Wiki

Maria Kozlowska - 5 years, 10 months ago

@Maria Kozlowska – Great $:)$

Sualeh Asif - 5 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$