Friday, August 9, 2013

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

Geometry Problem
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to see the complete problem 910.

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

2 comments:

  1. ∠KDJ
    = ∠KDA + ∠ADJ
    = ∠KDA + ∠ABJ
    = 1/2 [∠CDA + ∠ABC]
    = 90°

    ⇒ KJ is diameter of the circle O
    ⇒ OK = OJ = R

    In ΔOIK,
    IK² = R² + d² - 2Rd cos∠IOK

    In ΔOIJ,
    IJ² = R² + d² - 2Rd cos∠IOJ

    But ∠IOK + ∠IOJ = 180°,
    cos∠IOK = −cos∠IOJ

    Adding up, we have
    IK² + IJ² = 2 (R² + d²)

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  2. KI^2=R^2+d^2-2*R*d*cos<KOI, and IJ^2=R^2+d^2+2*R*d*cos<KOI. Adding the two equations gives the result.

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