If $A (x_{1}, y_{1}), B (x_{2}, y_{2}), C (x_{3}, y_{3})$ are the vertices of an equilateral triangle and $(x_{1} - 4)^{2} + (y_{1} - 5)^{2} = (x_{2} - 4)^{2} + (y_{2} - 5)^{2} = (x_{3} - 4)^{2} + (y_{3} - 5)^{2},$ then the value of $(y_{1} + y_{2} + y_{3}) - (x_{1} + x_{2} + x_{3})$ is

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Solution

Here, $△ A B C$ is equilateral.
$(x_{1} - 4)^{2} + (y_{1} - 5)^{2} = (x_{2} - 4)^{2} + (y_{2} - 5)^{2} = (x_{3} - 4)^{2} + (y_{3} - 5)^{2}$
$\Rightarrow A, B, C$ are equidistant from the point $(4, 5)$
$∴ (4, 5)$ is the circumcentre.
As it is an equilateral triangle, so the centroid coincides with circumcentre.
$(\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3}) = (4, 5) \Rightarrow x_{1} + x_{2} + x_{3} = 12, y_{1} + y_{2} + y_{3} = 15 ∴ (y_{1} + y_{2} + y_{3}) - (x_{1} + x_{2} + x_{3}) = 3$

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