Consider an equilateral triangle having vertices at the points A 2-3 e in - 2- - 2-3 e - iin - 6- - 2-3 e -i 5 - - 6 Let P be any point on itsincircle- then AP 2- BP 2- CP 2-

Let, nbsp; z_1 = nbsp; nbsp;2√(3) e^ iπ/2 , z_2= nbsp; nbsp;2√(3) e^ iπ/6 , z_3 = nbsp; nbsp;2√(3) e^ i5π/6 nbsp; Clearly, the points lie on ...

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Standard XII

Mathematics

Rotation Concept

Consider an e...

Question

Consider an equilateral triangle having vertices at the points $A (\frac{2}{\sqrt{3}} e^{i π / 2}), B (\frac{2}{\sqrt{3}} e^{- i π / 6}), C (\frac{2}{\sqrt{3}} e^{- i 5 π / 6})$ Let $P$ be any point on its incircle. then $A P^{2} + B P^{2} + C P^{2} =$

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Solution

Let,
$z_{1} = \frac{2}{\sqrt{3}} e^{i π / 2}, z_{2} = \frac{2}{\sqrt{3}} e^{- i π / 6}, z_{3} = \frac{2}{\sqrt{3}} e^{- i 5 π / 6}$
Clearly, the points lie on the circle $| z | = \frac{2}{\sqrt{3}}$ and $△ A B C$ is equilateral triangle.
Here, $z_{1} + z_{2} + z_{3} = 0 \Rightarrow_{1} +_{2} +_{3} = 0$
Since triangle is equilateral,
inradius $r = O D = \frac{1}{\sqrt{3}}$ unit
Let $P (z)$ be any point on the incircle, Now,
$A P^{2} = | z - z_{1} |^{2} = | z |^{2} + | z_{1} |^{2} - (z_{1} + ¯ ¯ ¯ z z_{1})$
Similarly,
$B P^{2} = | z |^{2} + | z_{2} |^{2} - (z_{2} + ¯ ¯ ¯ z z_{2})$
$C P^{2} = | z |^{2} + | z_{3} |^{2} - (z_{3} + ¯ ¯ ¯ z z_{3})$

$∴ A P^{2} + B P^{2} + C P^{2} = 3 | z |^{2} + | z_{1} |^{2} + | z_{2} |^{2} + | z_{3} |^{2} - z (_{1} +_{2} +_{3}) - ¯ ¯ ¯ z (z_{1} + z_{2} + z_{3})$
$= 5$

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