User:Kovac Jozef/sandbox

Trisect an arbitrary angle using only a straight edge and compass. Quandary that is as old as geometry

GEOMETRIC SOLUTION

Section One Trisecting 60-degree angle

Construction is as follows:
 * 1)   Draw an extended line. Mark the line as ‘a’
 * 2)  With compass’s needle on the line a draw an arbitrary half circle above the line, keep this compass setting and lock it if possible, for more uses.
 * 3)  Mark the half circle as ‘r’, mark center as point ‘O’ for reference in the construction following next.
 * 4)  Mark the intersection of line a and half circle r on right as point ‘A’.
 * 5)  Put the compass’s needle at the point A.
 * 6)  Draw an arc outward to the right so it intersects both, the line a and the half circle r.
 * 7)  Mark the new arc as ‘2r’ and the intersection of half circle r and new arc as point ‘B’.
 * 8)  Mark the intersection of arc 2r and the line a as point ‘R2’ indicating double radius from O.
 * 9)  Move the compass’ needle to point R2.
 * 10)  As in step #6 draw a third arc so it intersects both, the line a and the arc 2r.
 * 11)  Mark the third arc as ‘3r’.
 * 12)  Mark the intersection of line a and arc 3r as point ‘R3’ indicating triple radius from O.
 * 13)  With compass’ needle at R3, draw a half circle above the line.
 * 14)  Mark this half circle ‘4r’.
 * 15)  With compass’ needle back at O, adjust it to the width from point O to point R3, this compass setting is equal 3 times the original radius r.
 * 16)  Draw an arc long enough so it crosses from point R3 trough the half circle 4r.
 * 17)  Mark intersection of this large arc and arc 4r as the point ‘C’.
 * 18)  With the straight edge draw a line from the point O to the point B. This line forms 60-deg. angle AOB that was already created in steps #1 - #7. This is a standard construction in geometry.
 * 19)  With the straight edge draw another line from point O to the point C, marked in step #17. This line forms an angle AOC.
 * 20)  The new angle AOC is 20-deg. angle.

Trisected, the original 60-deg. angle AOB. Angle AOC is 20-deg. angle. Trisecting a 60-deg. angle was chosen for simplicity. As the first part of the construction is well known.

Section Two Trisecting an arbitrary angle of up to 180-degrees

Using the same technique, any arbitrary angle can be Trisected. Using a straight edge draw an angle. Draw the angle arms long. Mark the center of this angle O. With the compass’s needle at the point O, draw an arc r of small radius, which crosses both arms of the angle. Mark the lower arm intersection point A and upper arm intersection point B. making this the arbitrary angle AOB. Then using the compass extend this arc radius to 3 times r, on the lower arm. Mark the point R3. Put the compass’ needle back at the point O and adjust the compass to width from point O to point R3 and draw an arc. This arc’s radius is 3 times larger than the original radius. This arc must be long enough so it crosses both angle arms. Take the compass and put the needle at the point A. Then adjust the compass’ width from A to the point B. With compass adjusted to width from A to B, put the needle at the R3 point and draw an arc to cross the large (3 radiuses big) arc. Mark this intersection point C. Then using straight edge draw a new line from O to C. Thus constructed angle AOC is the Trisected angle AOB. To prove this use the compass’ last adjusted width AB and put he needle in the point C and draw an arc crossing the large arc again. Put the needle in this new intersection to draw another arc. This arc will cross the point where the upper angle arm and large arc intersect. This is the proof that the arc between A and B when transferred to the large (3 radiuses big) arc, fits in 3 times from point R3 to the upper arm. To trisect angles greater than 180-degrees, trisect the angle section that is over 180 and then ad 60-deg. to this angle. Example 210-deg. – 180 = 30-deg. Trisected equals 10-deg. Ad 60 + 10 = 70-deg. this is trisected 210-deg angle. More elaborate, but not a difficult construction using a straight edge and compass.

Section Three Fivesecting 60-degree angle

The same technique we described above of extending radius 3 times to trisect an angle can be used to create literally any angle section. Extending radius to multiple such as 5 will construct new angle that is 1/5 of an original angle. Radius can be multiplied by whole number, whole number and a fraction or just a fraction; any angle section can be constructed. Angle thus created, constructed using only straight edge and compass, and is accurate technically without limit.

MATHEMATICAL SOLUTION Simple mathematical proof of the geometrical solutions above.

Section One Proof Trisecting 60-degree angle Circumference L with unity radius r = 1 L = 2 x π x r = 2 x 3.14159 x 1 = 6.2831 Arc length of 60-deg. angle 360/6 = 60-deg. angle = L/6 = 6.2831/6 = 1.047197

Unity radius times 3, circumference L1 = 2 x π x 3r = 18.8495 Arc length 360/18 = 20-deg. angle = L1/18 = 18.8495/18 = 1.047197

Same as arc length of 60-deg. now 20-deg. with accuracy without limit.

Section Two Proof Trisecting an arbitrary angle of up to 180-degrees

Arbitrary angle 39-deg. Trisected 39/3 = 13-deg. Arc length (360/360)x39 = 39-deg. angle = (L/360)x39 = (6.2831/360)x39 = 0.6806

Unity radius times 3, circumference L1 = 2 x π x 3r = 18.8495 Arc length (360/360)x13 = 13-deg. angle = (L1/360)x13 = (18.8495/360)x13 = 0.6806

Same as arc length of 39-deg. now 13-deg. with accuracy without limit.

Section Three Proof Fivesecting 60-degree angle

A 60-deg. angle, Fivesected 60/5 = 12-deg. Arc length 360/6 = 60-deg. angle = L/6 = 6.283/6 = 1.047197

Unity radius times 5, circumference is L1 = 2 x π x 5r = 31.4159 Arc length 360/30 = 12-deg. angle = L1/30 = 31.4159/30 = 1.047197

Same as arc length of 60-deg. now 12-deg. with accuracy without limit.

Looking at the geometrical solution one can see other interesting angles that can be constructed further.