Proof that the equilateral triangle incentre and the circumcentre are the same point

Hello

I have a question on my maths assignment I need some help with.

The question says - Make a conjecture about the nature of a triangle where the Incentre and the Circumcentre are the same point. Prove your conjecture.

I know that its going to be an equilateral triangle, and I think I have to prove it by contradiction.

thanks

Re: Proof that the equilateral triangle incentre and the circumcentre are the same p

Incenter and circumcenter to coincide means that from the same point inside a triangle you can draw a circle that passes through each vertex and another smaller circle that is inscribed inside the triangle...draw these and prove that right angles generated by radius of inscribed circle on the sides of triangle cannot be right angles!

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Re: Proof that the equilateral triangle incentre and the circumcentre are the same p

Attachment 24796

Given $\displaystyle O$ is the Incenter and the Circumcenter of the $\displaystyle \Delta ABC$. Let $\displaystyle D,E,F$ be the foot of the perpendiculars from $\displaystyle O$ on sides $\displaystyle AB$, $\displaystyle BC$, $\displaystyle CA$ respectively. And $\displaystyle AD', BE', CF'$ be the angular bisectors of $\displaystyle \angle A$, $\displaystyle \angle B$, $\displaystyle \angle C$ respectively.

In $\displaystyle \Delta AOB$, $\displaystyle OA = OB =R$(circumradius), $\displaystyle \therefore \Delta OAB $ is isosceles $\displaystyle \therefore\angle OAB = \angle OBA \implies \frac{1}{2} \angle A = \frac{1}{2} \angle B $ (as $\displaystyle OA$ and $\displaystyle OB$ are angular bisectors also) $\displaystyle \imples \angle A = \angle B$

Similarly in $\displaystyle \Delta AOC$ we get $\displaystyle \angle C = \angle B$.

$\displaystyle \therefore \Delta ABC$ is equilateral triangle.