Monday, May 10, 2010

Problem 449: Congruent Tangent Circles, Diameter, Sagitta, Arc, Perpendicular, Chord

Geometry Problem
Click the figure below to see the complete problem 449 about Congruent Tangent Circles, Diameter, Sagitta, Arc, Perpendicular, Chord.

Problem 449: Congruent Tangent Circles, Diameter, Sagitta, Arc, Perpendicular, Chord.
See also:
Complete Problem 449
Level: High School, SAT Prep, College geometry

2 comments:

  1. Let O,I,R,r be center and radius of circumcircle and incircle of ABC respectively
    2[(R-a)+(R-b)]=2(OF+OD)=2(DB+FB)=AB+BC ⇒
    r=(AB+BC-AC)/2={2[(R-a)+(R-b)]-2R}/2=R-a-b ...(1)
    (R-a)²+(R-b)²=OF²+OD²=FD²=(AC/2)²=R² ⇒
    (r+a)²+(r+b)²=(r+a+b)² ⇔
    r=√(2ab) ...(2)
    consider similar triangles about incenter
    4x/(2R)=4x/AC=(r-x)/r ⇔
    x=rR/(R+2r)
    =r(a+b+r)/(a+b+3r) (by (1))
    =√(2ab)(a+b+√(2ab))/(a+b+3√(2ab)) (by (2))

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  2. Refer problems # 275 and 435 -- We've r^2=2ab, BC + 2a = AC, AB + 2b = AC & AC + 2r = AB + BC which leads to AC = 2(a+b+r). Further, 4x/AC = (r-x)/r or x = (r*AC)/(AC+4r)= 2r(a++b+r)/(2a+2b+6r) =(√(2ab)(a+b+√(2ab))/(a+b+3√(2ab)) QED.
    Ajit

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