Friday, May 7, 2010

Problem 447: Complete Quadrilateral, Inradii, Exradii, Sides.

Geometry Problem
Click the figure below to see the complete problem 447 about Complete Quadrilateral, Inradii, Exradii, Sides.

Problem 447: Complete Quadrilateral, Inradii, Exradii, Sides.
See also:
Complete Problem 447
Level: High School, SAT Prep, College geometry

2 comments:

  1. E, F, G, H tg points of AB, BC, CD, AD
    ▲EOaA ~ ▲AHOd, ▲EOaB ~ ▲BFOb (OaOd bisector of kites)
    =>
    AE/ra = AH/rd, EB/ra = BF/rb =>
    AE/ra + EB/ra = AH/rd + BF/rb

    a/ra = AH/rd + BF/rb (1)

    ▲FObC ~ ▲CGOc, ▲COcD ~ ▲DHOd
    FC/rb = CG/rc, GD/rc = DH/rd =>
    CG/rc + GD/rc = FC/rb + DH/rd

    c/rc = FC/rb + DH/rd (2)
    from (1) and (2)
    a/ra + c/rc = BF/rb + FC/rb + AH/rd + DH/rd

    A/ra + c/rc = b/rb + d/rd
    --------------------------------------------

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  2. Let external angle of ABCD at A,B,C,D be A',B',C',D'
    Let O be the excenter of ABE, excircle of ABE touch AB at P, then
    a/r_a=AP/r_a+PB/r_a=cot(A'/2)+cot(B'/2)
    similarly c/r_c=cot(C'/2)+cot(D'/2)
    and a/r_a+c/r_c=b/r_b+d/r_d=cot(A'/2)+cot(B'/2)+cot(C'/2)+cot(D'/2) is obvious

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