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Online Geometry Problem 907: Bicentric Quadrilateral, Incircle, Circumcircle, Circumscribed, Inscribed, Tangent, Inradius. Level: High School, Honors Geometry, College, Mathematics Education

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In a bicentric quadrilateral ABCD, AB, BC, CD, and AD are tangent to the incircle at E, F, G, and H. If AE = 3 and CG = 1, find the inradius.
A bicentric quadrilateral ABCD is a convex quadrilateral that has both an incircle I and a circumcircle O.
 

Geometry Problem 907: Bicentric Quadrilateral, Incircle, Circumcircle, Circumscribed, Inscribed, Tangent, Inradius
 

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 ill-structured 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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