I know that a triangle's mathematical center is not it's "visual" center, meaning it doesn't look good when aligned using the align tool, you have to use alternative methods. (Such as creating the triangle from the center and dragging outward proportionately)
Here's what puzzles me:
When I create the triangle from the center, and dragging out from the center, (You can see it's visually perfectly aligned and looks good). When I use the align tool it gets shifted over to that bad position.
Does that mean that holding down the Alt key doesn't actually properly align it to the center, and if so, is that a (better but possibly worse) fault in the drag from center tool (Hold down Alt key and drag)
Holding down Shift + Alt
Answer
Illustrator's align options use the bounding box as the area of definition, not the center point.
Therefore, when you click the align buttons, Illustrator is centering the triangle's bounding box within the circle's bounding box. It's not paying any attention to the actual center points.
The guides here represent the bounding boxes. (art shifts a bit when aligning, but they are the same size as you'll see at the end of the animation). 
In addition, when you draw holding the Option/Alt key down, you draw from area center - meaning everything moves outward equidistant from that center point. When you align, you align on visual center (which is what the bounding box center is). You'll notice the triangle's bounding box center is not actually centered in the triangle's area, it's visual centered based on shape edges.
I WISH there was some mechanism to align on actual centers in Illustrator, but pretty much all alignment options use the bounding box as reference. There's also no ability to show area center within Illustrator, only visual center.
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