Question

For an isosceles triangle, find its circumcentre (C), incentre (I), centroid
(G) and orthocentre (O). What do you observe? Are they collinear?

Answer 1

There's a concept of Euler Line, which says the centroid, orthocenter and the circumcenter lies on a straight line.

In case of an isosceles triangle, all of centroid, orthocentre, incentre and circumcentre lie on the same line.

We can find the centroid of an isosceles triangle from the point of intersection of the medians.

We can find the incentre of an isosceles triangle from the point of intersection of the angle bisectors.

We can find the circumcenter of an isosceles triangle from the point of intersection of perpendicular bisectors of the sides.