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Divide Into Four (Posted on 2010-12-02) Difficulty: 3 of 5
In ΔABC the altitude, angle bisector, and median from C divide ∠C into four equal angles.

What is the ratio of sides a and b (shorter:longer)?

See The Solution Submitted by Bractals    
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Solution Full solution | Comment 1 of 3
Call the places where the altitude, angle bisector and median intersect AB by D, E and F respectively.

angles BCD = DCE = ECF = FCA = x

h=the altitude CD
so we have the following lengths:
BD = h*tan x
ED = h*tan x
FD = h*tan 2x
AD = h*tan 3x
and since F bisects AB
h*tan(3x) - h*tan(2x) = h*tan(x) + h*tan(2x)
sin(3x)/cos(3x) - sin2x)/cos(2x) = sin(x)/cos(x) + sin(2x)/cos(2x)
cos(x)cos(2x)sin(3x) - cos(x)sin(2x)cos(3x) = sin(x)cos(2x)cos(3x) + cos(x)sin(2x)cos(3x)
cos(x)[cos(2x)sin(3x)-sin(2x)cos(3x)] = cos(3x)[sin(x)cos(2x)+cos(x)sin(2x)]
cos(x)sin(x) = cos(3x)sin(3x)
.5sin(2x) = .5sin(6x)
let y = 2x
sin(y) = sin(3y)
sin(y) = 3sin(y) - 4sin(y)^3
2sin(y) - 4sin(y)^3 = 0
1-2sin(y)^2 = 0
sin(y) = +/- sqrt(2)/2
y = 45 degrees = 2x
x = 22.5 degrees

So angle BCA = 4x = 90 degrees (a right triangle!)
angle BAC = 90-3x = 22.5 degrees

The ratio of a to b = BC/AC = tan(22.5) = tan(45/2) = (1-cos(45))/sin(45) = (1 - 1/sqrt(2))/(1/sqrt(2)) = sqrt(2) - 1
or about .4142
  Posted by Jer on 2010-12-02 15:34:48
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