Is the centroid always the centre of gravity for a triangle of uniform density?

[Suk-Sci]

Centroid...centre of gravity!

How to prove that centroid is the centre of gravity of a triangle...

 

[Hootenanny]

How to prove that centroid is the centre of gravity of a triangle...

In general, this isn't true. Indeed, it is true if the triangle in question has uniform density (or the density is uniformly distributed about the centroid).

In order to prove your statement for a triangle of uniform density, you merely need to compare the definition of the centre of mass and the definition of the centroid when the density is constant.