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

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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