Miquel Point of a Cyclic Complete Quadrilateral: Orthogonal Circles via a Midpoi

This example demonstrates a geometric configuration where, for a cyclic “complete quadrilateral” inside a circle, the Miquel point becomes the key to establishing an orthogonality (perpendicularity) relation between two circles. The construction highlights core concepts: Miquel point structures, midpoint-based circle construction, and the criterion that two circles are orthogonal, which can be confirmed by showing the relevant angles become right angles. In GeoGebra, you build a circle centered at the midpoint H of segment FG using Circle and Midpoint, then create the second circle (e.g., the original circle A). With Point, Line, Segment, and Intersect you determine the needed vertices and intersection points, while OrthogonalLine and Angle help verify the orthogonality/90° condition. Polygon is used to manage the quadrilateral’s vertex relations and Angle is used for the decisive angle check. Suitable for high-school advanced geometry and pre-university proof-oriented dynamic geometry lessons.