Preface

If you are unfamiliar with the common properties of complete quadrilaterals, you can first check it out on AOPS。

The Problem

Problem 18: As shown in the figure, A, B, C, D are four points arranged in order on ⊙O. Lines BA and CD intersect at point E; lines AC and BD intersect at point F. G and H are the midpoints of AC and BD respectively. Line GH intersects AB and CD at M and N. Construct the circumcircle ⊙P of △EMN, and construct ⊙Q with OF as the diameter. Prove: ⊙P and ⊙Q are tangent to each other.

By observation, points H and G lie on ⊙(OF), which can be proven by the Perpendicular Chord Theorem. When a problem involving tangent circles features a complete quadrilateral, the tangency point is often its Miquel point.

Introducing the Miquel point M of AEDCFB, we observe that M is the point of tangency. From the properties of complete quadrilaterals with concyclic feet, we can derive OM ⊥ EM (radical axis of constant difference, via inversion power of the blue circle), thus M lies on ⊙(OF).

Returning to the original problem, let T be the Miquel point of AEDCFB. Then HFGT are concyclic and AFTB are concyclic. Thus, T is also the Miquel point of AFGHMB, which makes MHTB concyclic. Similarly, NGTC are concyclic. Furthermore, we know that ∠MTH = ∠MBH = ∠GCN = ∠GTN. The proof follows from the properties of equal angles in tangent circles (refer to P17 for the detailed proof).

Summary

This is a very friendly problem; if you have just started learning about complete quadrilaterals, it serves as a good exercise. However, it might be a bit light for intensive competition training.

translated by me from my original post on bilibili
2026/3/4