For what kind of triangle, the centroid, circumcenter, and incenter is the same point?
Right-angled
Scalene
Equilateral
Isosceles
Solution
The centroid of a triangle is constructed by taking any given triangle and
connecting the midpoints of each leg of the triangle to the opposite
vertex.
The circumcenter is constructed by finding the midpoint of each leg of the
triangle and constructing a line perpendicular to that leg at its midpoint.
The in center is constructed by taking the intersection of the angle
bisectors of the three vertices of the triangle.
For an equilateral triangle, the angle bisectors connect to the midpoints
of each leg at 90°. Hence all these are one and the same.