[Math] 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