I'm trying to draw a circle (ellipse) inside a projected square in DraftSight, however the ellipse is not crossing the typical points (yellow circles)
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Some 2D CAD programs have ellipse drawing command "Ellipse Inscribed" or "By tangents". You point four tangential lines in your drawing and the program draws the only possible ellipse:
It works for arbitary lines, they can well be the sides of a parallelogram. If your program has ellipse drawing by 2 tangents and the centerpoint, you an use also it.
The screenshot is from LibreCAD (=freeware)
Another possibility is to draw a square and a circle inside it. Group them and apply three point linear transformation to the group. If the aspect ratio is unlocked, you can get skewing. Let the source points be three corners of the square and the target points be the corresponding points in the wanted parallellogram:
The screenshot is from a twelve years old version of TurboCAD.