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I wanted to center a square inside an triangle, but in certain attempts, I wasn't convinced that this is a good center for this type of triangle to position this square at.

So I am wondering how can I seek the good center point of it?

 

Here's the example.

So, I have a triangle, which I got the "centroid" out of it:

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However, I wanted the square to be more in the center of the space, I guess?

 

So, I noticed that the top is like a rotated square and got the center out of it, then positioned the square:

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However, now it was more at the top. What kind of center is this even? I don't think that this can be called "centroid" anymore, can it?

 

So what I did is, I measured the length inbetween the both centers and then made a point based on the half of the length:

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And now I thought, that the square would fit now more in a centered place.

 

But what I am wondering is. Did those "calculations" of those centers even make any sense? And is it even accurate?

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    I think the end result is good and that's really what matters. Perhaps someone could've rescued the same result by eye, but if geometry helped you reach the result why not? But I don't think you can ever use geometry as a proof that a design works. A simpler way of putting could be: centered vertically but nudged a bit down because the triangle is heavier at the bottom.
    – Wolff
    Sep 4 at 18:54
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    Perceived visual center can often be different than the mathematical center. Do you want a harmonious visual or a mathematically precise figure? If it's the visual... math be damned, use what you feel looks best.
    – Scott
    Sep 4 at 18:58
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    Its called optical balance. It is often the case that a human visual system does not consider mathematical teories correct.
    – joojaa
    Sep 4 at 20:46

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"Looks good" cannot be defined exactly with math. We know far too little of human mind. We can find afterwards math formulas which are actually arbitary, but happen to show that just the good looking version has got some balance.

In your case one well fitting math relation (=the imagined explanation) is:

"The square is nearly as far away from every side of the triangle when one measures the shortest distance."

Test it by comparing the result to others after making all three distances the same as accurately as possible.

BTW. Artificial intelligence developers try to put their programs to build math models of human taste by training the programs with huge numbers of empirical test results.

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