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Math Help - Proof that the equilateral triangle incentre and the circumcentre are the same point

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    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
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  2. #2
    Senior Member MaxJasper's Avatar
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    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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    Member kalyanram's Avatar
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    Re: Proof that the equilateral triangle incentre and the circumcentre are the same p

    Proof that the equilateral triangle  incentre and the circumcentre are the same point-incir.png
    Given O is the Incenter and the Circumcenter of the \Delta ABC. Let D,E,F be the foot of the perpendiculars from O on sides AB, BC, CA respectively. And AD', BE', CF' be the angular bisectors of \angle A, \angle B, \angle C respectively.

    In \Delta AOB, OA = OB =R(circumradius), \therefore \Delta OAB is isosceles  \therefore\angle OAB = \angle OBA \implies \frac{1}{2} \angle A = \frac{1}{2} \angle B (as OA and OB are angular bisectors also) \imples \angle A = \angle B
    Similarly in \Delta AOC we get \angle C = \angle B.

     \therefore \Delta ABC is equilateral triangle.
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