Is the circumcenter of an equilateral triangle equidistant from its 3 vertices? If yes, how can I prove that?
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asked Feb 2, 2014 at 5:21
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$\begingroup$ hotmath.com/hotmath_help/topics/circumcenter-theorem.html But have you really mean this as this proposition holds true for any triangle $\endgroup$– lab bhattacharjeeCommented Feb 2, 2014 at 5:24
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$\begingroup$ No, its okay. Thanks a lot! $\endgroup$– Anish BhattacharyaCommented Feb 2, 2014 at 5:28
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$\begingroup$ The circle and equilateral triangle both are symmetric figures.When circumcircle is drawn on equilateral triangle,the centre of the circle will be equidistant from 3 of it's vertices by symmetry. $\endgroup$– Devgeet PatelCommented Feb 2, 2014 at 5:43
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2 Answers
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Yes, that's true for any triangle.
The circumcircle of a triangle is a circle which passes through its vertices:
Since the centre of a cirle is equidistant from all point on the circle, it follows that the circumcentre is equidistant from the vertices.
In the case of an equilateral triangle, all the geometrical centres of the triange coincide. That means that not only the circumcentre, but the incentre, orthocentre and centroid are equidistant from the vertices too.
This isn't the case for other triangles.
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Yes. The circumcenter is the center of the circle passing through the vertices. So it is exactly at a distance $r$ away from the vertices. $r$ is the radius of the circumcircle.
