Friday, October 16, 2015

Geometry Problem 1154 Sangaku Problem: Three circles and a tangent line

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

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Online Math: Geometry Problem 1154 Sangaku Problem: Three circles and a tangent line.
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1 comment:

  1. Sumith PeirisOctober 16, 2015 at 9:36 AM

    Let the distance between the points of tangency of the tangent line be m between circles a and c and n between circles c and b so that the corresponding distance between a and b is m+n.

    We can write 3 equations using Pythagoras

    m^2 + (a-c)^2 = (a+c)^2 from which
    m^2 = 4ac.,,(1)
    Similarly
    n^2 = 4bc ....(2)
    (m+n)^2 = 4ab.....(3)

    Now eliminate m and n from these 3 equations

    4ac + 4bc + 8c sqrt(ab) = 4ab

    Divide by 4abc
    1/a + 1/b + 2/sqrt(ab) = 1/c
    Take the square root of both sides obviously ignoring the negative possibility

    1/sqrt a + 1/sqrt b = 1/ sqrt c

    Sumith Peiris
    Moratuwa
    Sri Lanka

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