Prove that the centroid, circumcenter, incenter, and orthocenter are collinear in an isosceles triangle

Question

I understand by the Euler line that the centroid, circumcenter, and orthocenter are collinear, but I don't know how to fit in the fact about the incenter and the isosceles triangle

Answer 1

• in $\bigtriangleup$ABC AB=AC
• WE take AD$\perp$BC.clearly BD=DC & $\angle$BAD=$\angle$DAC
• so clearly the circumcentre;orthocentre;incentre and centroid - all of them lie on the line AD
• SO that is prooved

Answer 2

Given the constructions of these centers $c_i$, a congruence $\Delta\to\Delta'$ of two triangles will transport $c_i$ to $c_i'\ $. As an isosceles triangle is congruent to its mirror image we have $c_i=c_i'$ for each of these centers. therefore they all lie on the symmetry axis.