We have a circular lamina and a circular wire, of the same radius. The centre of mass in both bodies is the centre of the circle, right? So far, no argument can go against this.
However, when we repeat the same process with a triangular lamina and a triangular wire frame, this breaks up and the centre of mass isn't the same point anymore. I want to imagine the frame as a lamina of an infinitely thin frame and vaccum in between.. this is supposed to be analogous to a uniform lamina.. My question is, why doesn't this hold? Let's imagine that I have a triangular lamina of metal and I kept cutting similar triangles from it, such that each similar triangle has its centre of mass coincident on the original triangle. I will eventually have a metal wire, right? The triangles, being similar and having their centre of masses coincident on each other, shouldn't have any effect on the position of the centre of mass, should it? Okay I know I'm wrong, I have verified this by calculation and drawing, but can anybody give me a satisfying mathematical or physical proof why I'm wrong? I feel the wrong part has to do with them being similar and having the c.m. coincident.. Something doesn't click here yet I can't put my hand on it. Thank you.
Here's how I've thought about the anology of cutting out ever larger triangles, and why it doesn't hold.
Consider the image below of a $3,4,5$ triangle with its medians shown. The intersection of the medians is the centre of mass of your lamina triangle.
Inside that are a blue and red triangle.
(For those affected by colour blindness, the red triangle is the one slightly further to the left.)
It's clear these are not the same, and hence the lamina triangle and wire frame triangle will have different centres of mass.