C172 Flight Instructor reported while in the pattern another aircraft was entering the pattern and not on frequency resulting in the other aircraft taking evasive action to avoid a collision.

Narrative

During an instrument stage check as the check instructor; I was entering the downwind at approximately midfield when a transient aircraft appeared on a collision course; attempting to enter the downwind at a 90-degree angle at midfield. They were considerably faster than us; and upon noticing the potential conflict; I took the flight controls; added full power to increase speed; and made a radio call stating that we were midfield right downwind. However; the other aircraft had not made any radio calls and proceeded to turn into the downwind right behind us. The G1000's projected flight path indicated that in 30-40 seconds; they could have collided with us. I returned control to the student once we were a little further from the aircraft and instructed them to turn base as soon as possible. The aircraft; still making no radio calls; then broke out of the pattern to the left and looped around to re-enter the downwind. They later performed a low pass over the runway with smoke on; executed a steep pitch-up maneuver; and then left the area. Throughout the entire incident; I did not hear a single radio call from them.