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Inscribed Circle (Posted on 2014-04-27) Difficulty: 3 of 5
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    
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Solution 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 equal
to its semiperimeter minus its hypotenuse,
   
    R = s - b                                   (4)

The line segment FE is the reflection of line
segment AD about the crease of the fold (the
perpendicular 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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