How do I turn this tri into a nice supporting edge quad?
And maybe how do I prevent this strange loop cuts?
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Select both adjacent faces (holding down Shift key) and make into one. Delete [X] > Limited Dissolve.
Initiate a loop-cut. [Ctrl + R] (PS: Don't press any mouse button until you would complete the following next step)
While you slide the loop, press [E]ven and you may need to press [F]lip as well to make the loop run parallel to the end you are sliding the edge to.