Move the sides of the triangle $ABC$ closer to the polygon, keeping them parallel to themselves, until they all touch the border of the polygon. If one of them intersects the border along a non-degenerate segment, we're done: that side was parallel to a side of the polygon to begin with. Otherwise, the new sides $BC,CA,AB$ meet the border of the polygon $\mathcal M$ in the three points $X,Y,Z$ respectively. Now start rotating the lines $BC,CA,AB$ at the same time, with equal angular velocities, around $X,Y,Z$ respectively. Eventually, we'll reach a position in which one of the lines will meet the border of the polygon along a non-degenerate segment. The triangle we've obtained now is similar to the initial one, but possibly smaller. Enlarge it (move its sides away from the polygon, keeping them parallel to themselves) until it reaches the initial size. This last triangle has the desired properties.

Nice proof

What？Possibly smaller?

You need to find a point to rotate,and the point is the intersection point of two of the vertical lines from $X,Y,Z$