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An example of harmonic division

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Let K_c and F_c be the feet of the symmedian and the altitude (both) from vertex C in triangle ABC. Then the following statements are equivalent:

  1. K_c and F_c divide AB harmonically
  2. the identity (b^2-a^2)^2=(ac)^2+(cb)^2 holds.


  • (Growing membership) In a non-right triangle ABC, let a,b,c be the side-lengths, h_a,h_b,h_c the altitudes, F_a,F_b, F_c the feet of the altitudes from the respective vertices, M_a,M_b,M_c the midpoints of sides BC,CA,AB in that order, E_a,E_b,E_c the Euler points, R the circumradius, O the circumcenter, N the nine-point center, H the orthocenter, O^c the reflection of O over side AB, O^b the reflection of O over side AC, O^a the reflection of O over side BC, K the symmedian point, K_c the foot of the symmedian from vertex C, and r the radius of the polar circle. PROVE that the following eighty-four statements are equivalent:
    1. E_c=F_c
    2. AH=b
    3. BH=a
    4. AE_a=\frac{b}{2}
    5. BE_b=\frac{a}{2}
    6. E_aM_c=\frac{a}{2}
    7. E_bM_c=\frac{b}{2}
    8. E_aM_b=h_c
    9. E_aF_b=\frac{AH}{2}
    10. E_bF_a=\frac{BH}{2}
    11. OO^a=b
    12. OO^b=a
    13. OM_a=\frac{b}{2}
    14. OM_b=\frac{a}{2}
    15. OM_c=h_c
    16. CH=2h_c
    17. h_a=AF_b
    18. h_b=BF_a
    19. AF_c=\frac{b^2}{2R}
    20. BF_c=\frac{a^2}{2R}
    21. \frac{a}{c} =\frac{h_c}{AF_b}
    22. \frac{b}{c}=\frac{h_c}{BF_a}
    23. \frac{a}{b}=\frac{BF_a}{AF_b}
    24. R=\frac{b^2-a^2}{2c}
    25. \frac{CK}{KK_c}=\left(\frac{a^2+b^2}{a^2-b^2}\right)^2
    26. r^2=\frac{1}{2}\left(a^2+b^2-c^2\right)
    27. h_c=R\cos C
    28. \cos A=\frac{b}{\sqrt{a^2+b^2}}
    29. \cos B=-\frac{a}{\sqrt{a^2+b^2}}
    30. \cos C=\frac{2ab}{a^2+b^2}
    31. \sin A=\frac{a}{\sqrt{a^2+b^2}}
    32. \sin B=\frac{b}{\sqrt{a^2+b^2}}
    33. \sin C=\frac{b^2-a^2}{a^2+b^2}
    34. \cos^2 A+\cos^2 B=1
    35. \sin^2 A+\sin^2 B=1
    36. a\cos A+b\cos B=0
    37. \sin A+\cos B=0
    38. \cos A-\sin B=0
    39. 2\cos A\cos B+\cos C=0
    40. 2\sin A\sin B-\cos C=0
    41. \cos A\cos B+\sin A\sin B=0
    42. a\cos A-b\cos B=\sqrt{a^2+b^2}\cos C
    43. \sin A\sin B=\frac{ab}{b^2-a^2}\sin C
    44. \sin^2B-\sin^2A=\sin C
    45. \cos^2A-\cos^2B=\sin C
    46. OH^2=5R^2-c^2
    47. h_a^2+h_b^2=AB^2
    48. \frac{h_a}{a}+\frac{h_b}{b}=\frac{c}{h_c}
    49. a^2+b^2=4R^2
    50. b=2R\cos A
    51. A-B=\pm 90^{\circ}
    52. (a^2-b^2)^2=(ac)^2+(cb)^2
    53. AH^2+BH^2+CH^2=8R^2-c^2
    54. \left(c+2AF_c\right)^2=a^2+b^2 or \left(c+2BF_c\right)^2=a^2+b^2
    55. K_c and F_c divide AB harmonically
    56. E_a is the reflection of M_b over side AB
    57. E_b is the reflection of M_a over side AB
    58. \triangle ABH is congruent to \triangle ABC
    59. \triangle OO^aO^b is congruent to \triangle ABC
    60. \triangle CNO is isosceles with CN=NO
    61. \triangle CNH is isosceles with CN=NH
    62. \triangle CHO is right angled at C
    63. N is the circumcenter of \triangle CHO
    64. \triangle O^cOC is right-angled at O
    65. \triangle O^cHC is right-angled at H
    66. quadrilateral O^cOHC is a rectangle
    67. the points O^c,O,C,H are concyclic with OH as diameter
    68. the reflection O^b of O over AC lies internally on AB
    69. the reflection O^a of O over BC lies externally on AB
    70. radius OC is parallel to side AB
    71. F_a is the reflection of F_b over side AB
    72. segment F_aF_b is perpendicular to side AB
    73. the nine-point center lies on AB
    74. the orthic triangle is isosceles with F_aF_c=F_bF_c
    75. the geometric mean theorem holds
    76. the bisector of \angle C has length l, where l^2=\frac{2a^2b^2}{a^2+b^2}
    77. the orthocenter is a reflection of vertex C over side AB
    78. segment HC is tangent to the circumcircle at point C
    79. median CM_c has the same length as the segment HM_c
    80. the bisector M_cO of AB is tangent to the nine-point circle at M_c
    81. AF_aBF_b is a convex kite with diagonals AB and F_aF_b
    82. altitude CF_c is tangent to the nine-point circle at F_c
    83. chord F_cM_c is a diameter of the nine-point circle
    84. segment HF_c is tangent to the nine-point circle at F_c.
      (Once we hit 100, we’ll stop this particular exercise.)