TRISECTING AN ANGLE

NOTE: all images are schematic and don't represent real values. They are just there to show the idea.

Problem: Find a general method for trisection of any given angle using the classical tools.

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 180o 'y' wont be intersected). Now draw a circle with the centre at the edge of the angle and with the radii 'z', let's call this circle 'c2':
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 180o:

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:
Then draw the circle with the radii 'd' whose centre is at the edge of the angle and then draw 'l2', all as described in pic7. Note that 'l2' now intersects 'y'(since it had to move away from the interior of the angle) and therefore you subtract this length between the edge of the angle and 'y' from 'd' to get a length 'z':
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.
Then draw 'c2' as in pic8 and connect the edge of the angle with the intersection points 'c2' and 'c1':
pic12)


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




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