Prove that if circum-circle and in-circle of a triangle are concentric, the triangle is an equalateral triangle.


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Question Number 33756 by Rasheed.Sindhi last updated on 23/Apr/18

$Prove that if circum - circle and$ $in - circle of a triangle are concentric,$ $the triangle is an equalateral triangle .$
	Answered by MJS last updated on 23/Apr/18

	$draw a circle that ‘ ‘ {sits}^{″} on O = (\begin{matrix} 0 \\ 0 \end{matrix}) with$ $r = radius of incircle; M_{1} = center of incircle$ $x^{2} + {(y - r)}^{2} = r^{2}; M_{1} = (\begin{matrix} 0 \\ r \end{matrix})$ $now put the longest side (let me call it c)$ $along the x - axis$ $A = (\begin{matrix} - p \\ 0 \end{matrix}); B = (\begin{matrix} q \\ 0 \end{matrix}) with p + q = c$ $M_{2} = center of the circumcircle$ $M_{2} = M_{1} \Rightarrow x_{M_{2}} = 0 \Rightarrow p = q = \frac{c}{2} \Rightarrow a = b$ $because if p \neq q then M_{2} will obviously$ $slip to one side$ ${what}^{'} s left to show is that y_{M_{2}} = r \Leftrightarrow a = b = c$ $this should be easy but I^{'} ve got to cook now; -)$
	Commented by Rasheed.Sindhi last updated on 23/Apr/18
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