We will discuss circumcentre and incentre of a triangle.
In general, the incentre and the circumcentre of a triangle are two distinct points.
Here in the triangle XYZ, the incentre is at P and the circumcentre is at O.
A special case: an equilateral triangle, the bisector of the opposite side, so it is also a median.
In the ∆XYZ, XP, YQ and ZR are the bisectors of ∠YXZ, ∠XYZ and ∠YZX respectively; they are also the perpendicular bisectors of YZ, ZX and XY respectively; they are also the medians of the triangle. So, their point of intersection, G, is the incentre, circumcentre as well as the centroid of the triangle. So, in an equilateral triangle, these three points are coincident.
If XY = YZ = ZX = 2a then in ∆XYP, YP = a and XP = \(\sqrt{3}\)a.
Now, XG = \(\frac{}{}\) = \(\frac{2}{3}\)XP = \(\frac{2\sqrt{3}a}{3}\), and GP = \(\frac{1}{3}\)XP = \(\frac{\sqrt{3}a}{3}\).
Therefore, radius of the circumcircle is XG = \(\frac{2\sqrt{3}a}{3}\) = \(\frac{2a}{\sqrt{3}}\) = \(\frac{Any side of the equilateral triangle}{\sqrt{3}}\).
The radius of the incircle = GP = \(\frac{a}{\sqrt{3}}\) = \(\frac{2a}{2\sqrt{3}}\) = \(\frac{Any side of the equilateral triangle}{2\sqrt{3}}\).
Therefore, radius of the circumcircle of an equilateral triangle = 2 × (Radius of the incircle).
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