MY 11 TRISECTION APPROXIMATION METHODS         Pg. 65

 

TRISECTION RELATIONSHIPS

 
tri12c.gif

FIG. 12

 

DESCRIPTION OF FIG. 12:

 
1. FIG. 12 is a quarter circle showing 3Æ and its sub angles Æ with all
    Pg. 66
  relevant horizontal and vertical lines displayed.
   
2. All distances are measured from the outside edge as indicated by
   
  the direction of the arrow in question, as shown in FIG. 12, above
   
3. Below in FIG. 13 path R1 can be described as follows:
 

DESCRIPTION OF FIG. 13:

 
1. FIG. 13 is a quarter circle showing 3Æ and its sub angles Æ with
   
  all relevant lines displayed, as shown below.
   
2. All distances represent the element it breaks as a unit or ended by
   
  an X at an intersect, as shown by FIG. 13 below.
   
3. FIG. 13, below, disproves NICOMEDES' theory for his
   
  CONCOID OF NICOMES' function:
   
  R1/R = 2/3 + N/(3cos3Æ) with 1 < R1/R < 2.
   
  It is really R1/R << 1; proof given below.
  Pg. 67

tri13c.gif
FIG. 13

  Pg. 68

DESCRIPTION OF FIG. 13: (cont'd)

 
3. R1 = ML + LO = 2a + (2m - 1) = 2a + (1 - 3a) = 1 - a
   
  R1 = ML + LO = (2/3)CL + (2m - 1)
   
  R1/CL = (2/3) + (n/cosÆ)/CL
   
  Since cos3Æ < cosÆ and CL < 3
   
  Thus R1/CL < (2/3) + n/(3cos3Æ) = 2/3 + 1/3 = 1
   
  But R > CL thus R1/R << R1/CL < 1   [my proof VOILA!]
   
  with CL ¹ 0, CL < 1, R = 1, and n = cos3Æ.
 
childrnc.gif
  Pg. 69

FORMULAS GENERATED:

 
1. FO/CO = FO/1 = GO/JO = HO/LO = cosÆ = m/q = n/(2m - 1)
   
2. HO = n = cosÆ(2m - 1) = m(2m - 1)/q
   
3. LO = (2m - 1) = n/cosÆ = cos3Æ/cosÆ = 1 - 4sin2Æ = 1 - 4p2
   
4. GO/DO = GO/1 = cos2Æ = m = cos2Æ - sin2Æ   [from CRC TABLES]
   
  GO = m = cos2Æ = 1 - 2sin2Æ = 1 - 2p2
   
  GO = m = cos2Æ = 2cos2Æ - 1 = 2m2/q2 - 1
   
5. HO/EO = HO/1 = cos3Æ = n = cosÆ(1 - 4sin2Æ)   [from CRC TABLES]
   
  HO = n = cos3Æ = cosÆ(4cos2Æ - 3) = (m/q)(4(m/q)2 - 3)
   
  n = cosÆ(2cos2Æ - 1) = (m/q)(2m - 1) = (1 - p2)½ (1 - 4p2)
   
6. FC/CO = FC/1 = sinÆ = p
   
7. GO = m = 1 - 2sin2Æ = 1 - 2p2 = qn/(2m - 1)
   
  or (2m - 1) = qn/m or 2m2 - m - qn = 0
   
  2p2 = 1 - m also 4p2 = (1 - m)2 + (2pm/q)2   [AD2 = AG2 + GD2 = CE2]
    Pg. 70
  (2m - 1) = 1 - 4p2
   
  (2m + 1) = 3 - 4p2
   
  (2m + 1)(2m - 1) = 4m2 - 1 = 3 - 16p2 + 16p4
   
  m2 = 1 - 4p2 + 4p4
   
  1 - m2 = 4p2(1 - p2)
   
  4p2 = (1 - m2)/(1 - p2) = (1 - m)(1 + m)/[(1 - p)(1 + p)]
   
8. HO = n = cosÆ(1 - 4sin2Æ) = (1 - p2)½(1 - 4p2) = m(1 - 4p2)/q
   
9. GD/DO = GD/1 = sin2Æ = 2sinÆcosÆ)   [from CRC TABLES]
   
  GD = sin2Æ = 2pn/(2m - 1) = 2pn/q = 2p(1 - p2)½
   
10. HE/EO = HE/1 = sin3Æ = sinÆ(3 - 4sin2Æ)   [from CRC TABLES]
   
  HE = sin3Æ = sinÆ(4cos2Æ - 1) = sinÆ(2cos2Æ + 1) = p(2m + 1)
   
  HE = p(4m/q - 1) = p(4n/(2m - 1) - 1) = p(3 - 4p2)
   
  (m + 1) = 2(1 - p2) = 2cos2Æ = 2m2/q2 = 2n2/(2m - 1)2
    Pg. 71
  (m - 1) = -2p2 = 2(cos2Æ - 1) = 2(m2/q2 - 1)
   
  (m - 1)(m + 1) = m2 - 1 = -4p2(1 - p2)
   
  cosÆ = n/(2m - 1) = (m + 1)½/2½ = (1 - p2)½ = (1 - sin2Æ)½
   
11. sin3Æ/sinÆ = 3 - 4sin2Æ = 4cos2Æ - 1   [from CRC TABLES]
   
  sin3Æ/sinÆ = 3 - 4p2 = 2m + 1 = 4(m/q)2 - 1
   
  2p = p(2m + 1) - p(2m - 1) = sin3Æ - tanÆcos3Æ
   
  (2m - 1) = n/cosÆ = cos3Æ/cosÆ = 1 - 4sin2Æ = 1 - 4p2
   
  4m = sin3Æ/sinÆ + cos3Æ/cosÆ = (2m + 1) + (2m - 1)
   
  2 = sin3Æ/sinÆ - cos3Æ/cosÆ = (2m + 1) - (2m - 1)
   
  See [MY KLAUSIAN TRI-FOCUS COSINE LAW on Pg. 39]
   
12. n = (2m - 1)cosÆ = (2m - 1)(1 - p2)½
   
  n = (1 - 4p2)(1 - p2)½
   
  n2 = 1 - 9p2 + 24p4 - 16p6
    Pg. 72
  2n2 = (m + 1)(2m - 1)2 = 4m3 - 3m + 1
   
  2n2 = 4(1 - 2p2)3 - 3(1 - 2p2) + 1
   
13. [PROPERTY OF A CIRCLE: any vertical segment, above the
   
  diameter, squared equals the product of the left and right segments
   
  on the diameter - THEOREM.]
   
    i. p2 = (1 - m/q)(m/q + 1) = (1 - m2/q2)   [FC2 = AF(FO + 1)]
   
   ii. (2pm/q)2 = (1 - m)(m + 1) = (1 - m2)   [GD2 = AG(GO + 1)]
   
  iii. p2(2m + 1)2 = (1 - n)(n + 1) = 1 - n2   [HE2 = AH(HO + 1)]
   
14. HE = HL + LE = HL + p(2m - 1) = sin3Æ
   
  HE = p(2m + 1) - p(2m - 1) = 2p
   
15. Let FH = 3e = cosÆ - n = n/(2m - 1) - n = 2n(1 - m)/(2m - 1)
   
  FH = 3e = cosÆ - n = (1 - p2)½ - (1 - 4p2)(1 - p2)½
   
  FH = 3e = 4p2(1 - p2)½ = 2sinÆsin2Æ, see DEFH above
    Pg. 73
  1 - m2 = 4p2(1 - p2) = [(3e)/(2p)]2 = 2.25(e/p)2   [a UNIT CIRCLE]
   
16. Let CL = 3a = 1 - (2m - 1) = 2(1 - m) = 4p2, see DCFO above
   
  Let CM = CL/3 = a = e/cosÆ = eq/m = 4p2/3
   
17. Let FE = 3b and 9b2 = (m/q - n)2 + (p(2m + 1))2   [FE2 = FH2 + HE2]
   
  see DEFH above.
   
18. Let HE = 3h = p(2m + 1) = sin3Æ = p(1 - 4p2)
   
  Let QM = HE/3 = h = p(2m + 1)/3 = p(1 - 4p2)/3
   
19. [ TRIANGLE RATIO LAW]
   
  In DGJO:
   
  GJ/HL : GO/HO : JO/LO : pq/(p(2m - 1)) : m/n = q/(2m - 1)
   
  In DGDO:
   
  GD/HK : GO/HO : DO/KO : (2pm/q)/(2pn/q) : m/n = 1/(n/m)
    Pg. 74
  In similar DFCM & DMEL:
   
  ME/FM = ML/CM = LE/FC = 2b/b = 2a/a = 2p/p = 2
   
  In DEFH:
   
  FE/FM = FH/FQ = HE/QM = 3b/b = 3e/e = 3h/h = 3
   
  In DPFO & DCFO:
   
  OP/QM = FO/FQ = s/h = (n + 3e)/e or OP = s = h(n + 3e)/e
   
  1/QM = 1/FC + 1/OP = 1/h = 1/p + 1/s   [from FOCUS FORMULA]
   
  or OP = s = ph/(p - h) = p(p(3 - 4p2)/3)/(p - (p(3 - 4p2)/3))
   
  s/h = (n + 3e)/e = (1 + p2)½/e or s = (3 + n/e)h
   
  n/e = s/h - 3 = 1/(1 - h/p) - 3
   
  n/e = (3h - 2p)/(p - h) = (1 - 4p2)(1 - p2)½/ (4p2(1 - p2)½/3)
   
  n/e = 3(1 - 4p2)/(4p2) = .75p-2(1 - 4p2)
   
  or OP = s = (2p + p(2m - 1))(1 + p2)½/((1 - 4p2)(1 - p2)½)
    Pg. 75
  s = (2p + p(2m - 1))/(1 - 4p2) = (2p + p(1 - 4p2))/(1 - 4p2)
   
  p/h = (3e + n)/(2e + n) = cosÆ/(2e + n)= (3 + n/e)/(2 + n/e)
   
20. (3b)2 = p2(2m + 1)2 + (3e)2 = p2(3 - 4p2)2 + (4p2(1 - p2)½)2
   
  (3b)2 = p2(3 - 4p2)2 + 16p4(1 - p2) = 9p2 - 8p4 = p2(9 - 8p2)
   
  b = p(9 - 8p2)½/3 = p(4m + 5)½/3
   
21. Let AF = x = 1 - cosÆ = 1 - m/q = n/(2m - 1) = 1 - (1 - p2)½
   
22. Let QG = y = 2e + n - m = 4sinÆsin2Æ/3 + cos3Æ - (1 - 2sin2Æ)
   
  QG = y = 2(4p2(1 - p2)½)/3 + (1 - 4p2)(1 - p2)½ - (1 - 2p2)
   
  y = (1 - p2)½)(8p2/3 + 1 - 4p2) - (1 - 2p2)
   
23. FG = FQ + QG = e + y = cosÆ - cos2Æ
   
  FG = m/q - m = (1 - p2)½ - (1 - 2p2)
   
24. ÐFEH = b = sin-1(e/b) = sin-14p[(1 - p2)/(9 - 8p2)½]
   
  b = tan-1(e/h) = tan-14p[(1 - p2)/(1 - 4p2)½]
    Pg. 76
  b = tan-1(e/h) = tan-1(2sin2ÆsinÆ/sin3Æ)
   
  b = cos-1(h/b) = cos-1[(3 - 4p2)/(9 - 8p2)½]
   
  a2 = p2 + b2 - 2pbcosb   [COSINE LAW]
   
  b = cos-1[(p2 + b2 - a2)/(2pb)]
   
  b = cos-1(p2 + p2(9 - 8p2)/3 - 16p4/3)
   
  b = cos-1(4p2(1 - 2p2))
   
  b = cos-1(2m(1 - m))
   
  e2 = b2 + h2 - 2bhcosb   [COSINE LAW]
   
  b = cos-1[(b2 + h2 - e2)/(2bh)]
   
  b = cos-1[(p2(9 - 8p2)/9 + p2(1 - 4p2)2/9 - 16p4(1 - p2)/9)/(2(p(9 - 8p2)½/3)(p(1 - 4p2)/3))]
   
  b = cos-1[p(9 - 32p2 + 32p4)/ (6(9 - 8p2)½ (1 - 4p2))]
   
  tanb = 1 - [1/(2m(1 - m))]½ = (3e + n)/s
   
  tanb = e/h = 3e/(p(2m + 1))
    Pg. 77
25. Let BT = z = 1 - 3h = 1 - p(3 - 4p2)½ = 1 - p(2m + 1)
   
26. In similar DCFM & DMPO:
   
  s/p = (2a + 2m - 1)/a = (2b + EP)/b = (3h + TP)/p
   
  EP = b(2m - 1)/a = p(9 - 8p2)½(1 - 4p2)/(4p2/3)
   
  EP = .75(9 - 8p2)½(1 - 4p2)/p and TP=s - 3h
   
27. In DFPO & DEPT:
   
  sq/m = TP/n or TP = sqn/m thus s - 3h = sqn/m
   
  s = 3h/(1 - qn/m) = 3h/(1 - 2m + 1) = 3h/(2(1 - m))
   
  s = .75h/p2 = (2m + 1)/(4p) = (3 - 4p2)/(4p) = (.75/p) - p
   
  TP = s - 3h = 3h/4p2 - 3h = 3h(1 - 4p2)/(4p2)
   
  TP = (3 - 4p2)(1 - 4p2)/(4p) = s(1 - 4p2)
   
  TP = (1 - 13p2 + 16p4)/(4p)
   
28. 3e = cosÆ - n = n/(2m - 1) - n = n(2 - 2m)/(2m - 1)
    Pg. 77a
28. 3e = 2sinÆsin2Æ = 3sin2ÆcosÆ = 4p2(1 - p2)½
   
  (3e)2 = (3b)2 - (3h)2
   
29. 3h = sin3Æ = sinÆ(2sin2Æ + 1) = p(2m + 1) = p(3 - 4p2)
   
  (3h)2 = (3b)2 - (3e)2
   
30. (3b)2 = p2(2m + 1)2 + (n/(2m - 1) - n)2 = 9p2 - 8p4
   
  3b = p(9 - 8p2)½ = p(4m + 5)½
   
  (3b)2 = (3h)2 + (3e)2
   
31. 4p2 = 4sin2Æ = 1 - cos3Æ/cosÆ = 3 - sin3Æ/sinÆ
   
  4p2 = 4(1 + n2 + 24p4 - 16p6)/9
   
  4p2 = 2(1 - m) = 1 - (2m - 1) = (1 - m2)/(1 - p2)
   
  4p2 = 1 - n/cosÆ = 1 - qn/m
 

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