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 Inscribed Circle (Posted on 2014-04-27)
A square sheet of paper ABCD is folded with D falling on E along BC, A falling on F and EF intersecting AB at G. A circle is inscribed in triangle GBE with radius R .

Determine |FG| in terms of R.

 No Solution Yet Submitted by K Sengupta No Rating

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 Solution | Comment 1 of 2
`Let    a = |FG|    b = |EG|    e = |BG|    f = |AG|    g = |BE|    s = the semiperimeter of triangle GBE,    t = the length of the square's side, and    x = the measure of /ADE.`
`    s = (b + e + g)/2                           (1)      t = |AB| = |BC| = |CD| = |DA|               (2)`
`Applying the Pythagorean Theorem to triangle GBE,`
`    b^2 = e^2 + g^2                             (3) `
`The inradius of right triangle GBE is equalto its semiperimeter minus its hypotenuse,       R = s - b                                   (4)`
`The line segment FE is the reflection of linesegment AD about the crease of the fold (theperpendicular bisector of line segment DE),`
`    t = |AD| = |FE| = |FG| + |EG| = a + b       (5)`
`ADEF is therefore an isosceles trapezoid,`
`    /FED = /ADE = x  and  /AFE = /FAD = 180-x   (6)`
`Equations (6) imply`
`    /CED = x  and  /FAG = 90-x                  (7)`
`Applying the law of sines to triangle AFG,`
`         a             f    ----------- = -----------     sin(/FAG)     sin(/AFG)`
`       t - b         t - e    ----------- = ------------     sin(90-x)     sin(180-x)`
`       t - b         t - e    ----------- = ------------                  (8)                      cos(x)        sin(x)`
`Therefore,`
`     t - e                          |CD|                          `
`        ------- = tan(x) = tan(/CED) =      ------`
`     t - b                          |CE|                     t            = -------                           (9)               t - g`
`Combining equations (1),(3)-(5), and (9) gives`
`    |FG| = R`
`QED`

 Posted by Bractals on 2014-04-27 12:54:16

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