A bicentric polygon is one which possesses both an incircle and a circumcircle. To every bicentric polygon belong sets of circles which relate to them in the way described below (Figure 1). We take the triangle as our exemplar. We state results which students can verify using elementary methods. They may consult [1] for our own proofs but in several cases they may spot a neater approach. If so they should submit it to this journal as Feedback.