Think something different

Geometry Level 5

$\begin{cases} { x }^{ 2 } + \dfrac { { y }^{ 2 } }{ 3 } + xy = 25 \\ \dfrac { { y }^{ 2 } }{ 3 } + { z }^{ 2 } = 9 \\ { z }^{ 2 } + {x }^{ 2 } + zx = 16\end{cases}$

Positive reals $x$ , $y$ and $z$ satisfy the system of equations above. Find $\sqrt 3(xy + 2yz + 3zx)$ .

The answer is 72.

1 solution

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Jun 26, 2020

Consider a triangle $ABC$ with $AB=3,BC=4,CA=5$ .

$P$ is in $\triangle ABC$ such that $PA=\frac{y}{\sqrt{3}},PB=z,PC=x$

and $\angle APB=90^\circ,\angle BPC=120^\circ, \angle CPA=150^\circ$ .

Then the expression becomes twelve times the area of $\triangle ABC$ .

Think something different

The answer is 72.

1 solution

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