Sunday, March 15, 2015

Problem 1099 Four Circles, Common External Tangent, Common Internal Tangent, Radius, Metric Relations, Sangaku

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

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Online Math: Geometry Problem 1099: Four Circles, Common External Tangent, Common Internal Tangent, Radius, Metric Relations, Sangaku.

3 comments:

  1. http://s2.postimg.org/3z5fe947d/pro_1099.png

    Draw lines per attached sketch
    Let CA1 cut DB1 at A . We have O1 and O2 are incircle and excircle of triangle ADC
    We have DE= CF … ( property of incircle and excicle of a triangle ADC)
    DE= r2.tan(D/2)
    CF= r1/tan(C/2)
    DE=CF => r2= r1/(tan(D/2). tan(C/2)) ….. (1)
    In right triangle O1HO4 we have HO4 ^2= (r1+r4)^2 – (r1-r4)^2 => HO4=2.sqrt(r1.r4)
    So tan(D/2)= (r1-r4)/(2.sqrt(r1.r4))
    Similarly tan(C/2) = (r1-r3)/(2.sqrt(r1.r3))
    Replace value of tan(C/2) and tan(D/2) in (1) we have
    r2= 4. r1^2.sqrt(r3-r4)/((r1-r3).(r1-r4))

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  2. There is a typo error in the last line of my comment. The correction will be
    r2= 4. r1^2.sqrt(r3.r4)/((r1-r3).(r1-r4))

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  3. Purely geometry proof


    Let A1A2 be tangent to circle O3 at A3 and let CD be tangent to circles O1,
    O2, O3, O4 at C1,C2,C3,C4. Let BD be tangent to circle O4 at B4.


    Further let A1C=CC1= a, A3C = C3C =p, C1C2 = b, C2C4 = c, C4D = d so that
    B4D = d and so B1B4 = b+c and DB2 =c+d


    Now O1A1C and O2A2C are similar triangles so r1r2 = a(a+b)…..(1)


    Similarly r1r2 = (b+c+d)(c+d) …..(2)


    From (1) and (2) a=c+d …(3)


    From similar triangles a/p =r1/r3 and using Pythagoras (a-p)^2 = (r1+r3)^2
    – (r1-r3)^2 = 4r1r3 , hence solving and simplifying,


    a=2r1sqrt(r1r3)/(r1-r3)….(4) and similarly from (3)


    C1D = b+c+d = a+b = 2r1sqrt(r1r4)/(r1-r4)…(5)


    From (4) and (5), a(a+b) = 4r1^3.sqrt(r3r4)/{(r1-r3)(r1-r4)}


    Hence from (1) r2 = a(a+b)/r1 = 4r1^2.sqrt(r3r4)/{(r1-r3)(r1-r4)}


    Sumith Peiris
    Moratuwa
    Sri Lanka

    ReplyDelete