MY 11 TRISECTION APPROXIMATION METHODS         Pg. 50

 

TRISECTION CHART

 
tri10c.gif


FIG. 10

 

CONSTRUCTION:

 
1. Draw any suitable straight line OBA and draw a unit circle GB
   
  with center pt O and a suitable radius OB = R = 1 unit, as shown
   
  in FIG. 10 above.
    Pg. 51
2. Draw any straight line OC so that ÐCOA = Æ with pt C lying on the
   
  circumference of the previous unity circle, see FIG. 10 above.
   
3. Similarly draw a straight line OD so that the ÐDOA = 3Æ
   
  (measured counter clockwise) with pt D lying on the circumference
   
  of the previous unity circle BCD, as shown in FIG. 10 above.
   
4. Join pts C & D with a straight line.
   
5. Draw a circle with center pt O and radius of length CD to intersect
   
  the line OD extended at pt E, as shown in FIG. 10 above.
   
6. Repeat all steps 1 to 5 for every pt E' and ÐE'OA = 3Æ' and
   
  ÐC'OA = Æ' from 0o to 360o to get the chart or complex figure
   
  GAHEO shown in FIG. 10 above.
   
 

NOTE:

   
  Line GOH is the Y-axis and line OBA is the X-axis.
  Pg. 52

METHOD OF TRISECTION:

 
1. By going backwards from 3Æ' to get Æ' one can trisect all angles
   
  with little error.
   
2. Assume in FIG. 10 that the angle to be trisected is ÐEOB = 3b.
   
3. With center D and radius length OE draw an arc to intersect the
   
  inner unity circle CGDRB at pt C.
   
4. Join pts C & O with a straight line.
   
5. Thus ÐCOB = b trisects ÐDOA = 3b as the required.
 
carl.gif
  Pg. 53

PROOF:

 
1. In DCOD:
   
  Let ÐEOA = 3b and ÐCOE = Æ
   
  Since 2 sides OD = OC = R therefore DCOD is isosceles.
   
  length EO = DC = 2(OC)sin(ÐEOA - ÐCOA) =
   
  = 2RsinÐCOD = 2Rsin(3b - Æ)
   
2. But length EO = 2Rsin2Æ, construction
   
  Thus (3b - Æ) = 2Æ or 3b = 3Æ and b = Æ.
   
3. Thus ÐCOA = ÐEDA/3 = 3b/3 = b
   
  Thus ÐCOA = b trisects ÐEOA = 3b, as required.
 
cones4.gif

UPSIDE DOWN CAKE
 

PREV TOP NEXT
MAIN MENU HOME PAGE
TRISECTION MENU 0 1 2 3 6 8 [10] 11 12 14 15 16 17 27 28
Hosted by www.Geocities.ws

1