You start out with an angle:

pic1) |

Draw a circle with the angle in its centre and draw to radii's following the sides of the angle. You can give the radii of the circle any length you want; this is completely arbitrary. The radii of this circle we call 'r' and the circles diameter we call 'd':

pic2) Our original angle in white. We drew a circle around it(red)
and the circles radii's are the green lines going out and following the
sides of the angle. |

You can then erase the circle and connect the tips of the radii lines:

pic3) The green radii's are connected by the red line. Let's call
this red line 'y' |

Then split 'y' in two parts by drawing two arbitrarily large circles that intersects each other. One whose centre is allocated on the tip of one of the green radii lines and another whose centre lies on the other green radii line. Draw a line that goes through the two points where the two new circles intersects and you've split 'y' in two:

pic4) The blue circles are the ones that is used for splitting 'y',
they have their centres at the tips of the radii lines, and the yellow
line is the line that goes through the circle's intersection points and
splits 'y.' |

We draw a circle whose centre is at the middle of 'y' and that has a radii of 'd'(that is it's radii is the same length as the diameter of our original circle in pic2). Then draw a line that goes through the edge of the angle and through 'y's middle point and on to intersect the newly draw circle. This line may not leave the angle area, we call this line 'l1':

pic5) The blue circle has a radii of 'd' and it's centre is
allocated on the middle of 'y'(the place where the yellow and red line
intersects). The yellow line is 'l1' and goes from the angles edge, at the
bottom, through 'y' and intersects the blue circle inside the
angle(this will be important further down). |

At the place where this new line intersects the circle with radii 'd' you draw another circle with the radii 'r'(that is the same radii as the circle in pic2), let's call this circle 'c1':

pic6) The purple circle is 'c1'and has it's centre where the blue
circle and the yellow line intersects. |

Then draw another circle with a diameter 'd' that has it's centre at the angles edge. After that you draw a straight line that starts from the edge of the angle and continues away from the interior of the angle and intersects the newly draw circle with the radii 'd', we call this line 'l2':

pic7) The purple circle is 'c1', which we drew previously, and the
yellow circle is our new circle with the radii 'd', it has it's centre at
the edge i.e. at the place where the two green lines meet. The blue line
is 'l2'. |

See if 'l1' intersects 'y'. If it does you subtract the length from 'y' and to the edge of the angle from 'd'(not mathematically though, you have to do it graphically by accommodating your compass to the space between the line 'y' and the intersection point on the yellow circle) , we call this new length 'z'. If 'l1' doesn't intersect 'y' we're left with the whole of the diameter(i.e. 'z'='d' and for all angles below 180

pic8) The purple circle is 'c1', which we drew previously, and the
blue circle is our new circle with the radii 'z', the circle we call 'c2'.
Note that since 'l2' didn't intersect 'y' this new, blue, circle is
identical to the yellow circle in pic7. |

We draw two lines from the edge of the angle to both the places where 'c1' and 'c2' intersects and then we've split the angle in three(though we really haven't):

pic9) The purple circle is 'c1' and the blue circle is 'c2'. The
two yellow lines cut's the angle in three |

Now I'll also show how to do it for angle over 180

Draw and split 'y' as described in pic3-4 and draw 'c1' as in pic5-6.

pic10) Note that since the line 'l1' had to go through both the
edge of the angel and the middle of 'y' and intersect the circle
inside the angle area, the intersection point, and therefore 'c1',
gets much closer to the edge of the angle: |

pic11) The purple line between the edge of the angle and 'y' is the
part we've subtracted from 'l2', the yellow line above is what we're left
with. |

pic12) |

Why it doesn't work: Uuumm... It simply doesn't work? I might be a fairly good approximation.

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