For, if it cuts them, the straight line drawn at right angles to the diameter of the circle from its end will be found to fall within the circle, which was proved absurd, therefore the circle described with center D and radius one of the straight lines DE, DF, or DG does not cut the straight linesAB, BC, and CA Therefore it touches them, and is the circle inscribed in the triangle ABC.
Let it be inscribed as FGE.