General aviation instructor reported a near miss with another aircraft while in the traffic patter at a non-towered airport during a training flight. The instructor maneuvered away from the other aircraft on the downwind leg to provide separation.

Narrative

Instructor and student were practicing a single engine traffic patterns at buckeye. While in the upwind a call was made that Aircraft Y was about 5 miles to the south and would be entering the pattern. Aware of the traffic we continued our single engine pattern and before long the traffic was entering in the pattern. There was still a fair distance between us and the other aircraft; but by the time we were midfield in the downwind; our aircraft was notifying us that Aircraft Y was within less then 1 mile of us. Not thinking much of it; before we realized it; the other traffic had come within a range of 400-800 ft horizontally of us. We called the other traffic on the CTAF asking if they had us in sight. The Aircraft Y confirmed they had us in sight but by this point; on the traffic map; it was showing they were right over us with a 100 ft vertical clearance. In such a high load situation while being single engine in the traffic pattern we didn't think to make a call to ask for spacing or anything along those lines. We were more worried about scanning and seeing if we could find the other traffic. Looking out the window the traffic was just behind us to the left at our 45. It was close enough that we were able to make out noticeable details of the plane; color; shape; etc. We continued our pattern and turned away from them as soon as we could but their ground speed was slightly faster then ours. Luckily there was no actual air collision to note; but it was extremely close. We were able to avoid any dangerous situations but nonetheless it was an extremely close and stress inducing moment.