Problem 675: Quadrilateral, 45 and 90 Degrees,
Circumcircles, Intersecting Circles, Secant, Collinear Points.
Level: High School, College, Math Education.
The drawing shows the quadrilateral ABCD with the exterior angles B and D equal to 45 degrees. Point E is on AB extended, such that angle DCE = 90 degrees, and point F is on AD extended, such that angle BCF = 90 degrees. CIrcumcircle O of triangle ABD and circumcircle O' of triangle AEF meet at G. CG meets circle O at H. Prove that points A, O, and H are collinear.
Geometry problem solving
Geometry problem solving is one of the most challenging skills for students to learn. When a problem requires auxiliary construction, the difficulty of the problem increases drastically, perhaps because deciding which construction to make is an illstructured problem. By “construction,” we mean adding geometric figures (points, lines, planes) to a problem figure that wasn’t mentioned as "given."

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