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Online Geometry Problem 910: Bicentric Quadrilateral, Distance between the Incenter and Circumcenter, Incircle, Circumcircle, Circumscribed, Inscribed, Circumradius. Level: High School, College, Mathematics Education

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In a bicentric quadrilateral ABCD, I is the incenter, O the circumcenter, R the circumradius, and IO = d. BI and DI extended meet the circumcircle O at J and K, respectively. Prove that IK2 + IJ2 = 2(R2 + d2).

A bicentric quadrilateral ABCD is a convex quadrilateral that has both an incircle I and a circumcircle O.
 

Geometry Problem 910: Bicentric Quadrilateral, Distance between the Incenter and Circumcenter, Incircle, Circumcircle, Circumscribed, Inscribed, Circumradius
 


 

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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