1 thought on “Two Curious Semicircles

  1. SANJAY GODSE

    Sir,
    I have found one more simple solution. With reference to your diagram, let the blue segment between the center of the circle and the touching point of the two semicircles be x.
    Then ^2 = r^2 + (r+x)^2 and
    ^2 = R^2 + (R-x)^2

    Equating the two RHSs,

    r^2 + r^2 + x^2 + 2*r*x = R^2 + R^2 + x^2 – 2*R*x

    R^2 – r^2 = x*(R+r)

    (R+r)*(R-r) = x*(R+r)

    x = R-r

    Substituting in first or second equation,

    ^2 = r^2 + R^2

    Rest follows.

    Reply

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