Let A , B , C be equilateral triangle with 2 3 A = e iπ /2 , 2 3 B = e − iπ /6 , 2 3 C = e − i 5 π /6 . Let P be any point on the incricle of △ A BC . Find the value of P A 2 + P B 2 + P C

Question

Let A , B , C be equilateral triangle with 2 3 ​ ​ A = e iπ /2 , 2 3 ​ ​ B = e − iπ /6 , 2 3 ​ ​ C = e − i 5 π /6 . Let P be any point on the incricle of △ A BC . Find the value of P A 2 + P B 2 + P C 2

TestHub · Accepted Answer

Given triangle is an equilateral triangle

∴ incircle is $x^{2} + y^{2} = \frac{1}{3}$

Let point on the in circle is $(x, y)$

$∴ P A^{2} + P B^{2} + P C^{2} = x^{2} + (y - \frac{2}{3})^{2} + (x - 1)^{2} + (y + \frac{1}{3})^{2} + (x + 1)^{2} + (y + \frac{1}{3}) 2$

$= 3 (x^{2} + y^{2}) + 4$

$= 1 + 4 = 5$

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