You need the angle bisector of all three points of the triangle, the intersection of those is called the incentre, and that will be the center of your inscribed circle.
Then, measure with a ruler the distance between the incentre and the side, and make interim marks that same distance away from the incentre, then you can trace it.
OTOH if your circle doesn't need to touch the sides, then just take a coin small enough and trace it in your triangle lol.
This might be what you want:
https://en.m.wikipedia.org/wiki/Poncelet%E2%80%93Steiner_theorem
Specifically this part:
it is naturally understood that circles and their arcs cannot be drawn without the compass
Some straight-line constructions, you might or might not need a compass to locate properly in order to draw them. But as the tools are usually defined for these kinds of constructions, actually drawing the circle without a compass is not possible.
Well. If you can find the center of the inscribed circle and its tangent points on the triangle, that might be what OP wanted.
This question is too vague at best. Is the circle supposed to be tangent to all sides of the triangle? Why no compass? Drawing freehand, pencil and paper?