{
  "id": "enwiki-001363840823",
  "slug": "conway-triangle-notation",
  "title": "Conway triangle notation",
  "qid": "",
  "lead": "In geometry, the Conway triangle notation simplifies and clarifies the algebraic expression of various trigonometric relationships in a triangle. Using the symbol for twice the triangle's area, the symbol is defined to mean times the cotangent of any arbitrary angle . The notation is named after English mathematician John Horton Conway, who promoted its use, but essentially the same notation (using instead of ) can be found in an 1894 paper by Spanish mathematician .",
  "body": "In geometry, the Conway triangle notation simplifies and clarifies the algebraic expression of various trigonometric relationships in a triangle. Using the symbol for twice the triangle's area, the symbol is defined to mean times the cotangent of any arbitrary angle . The notation is named after English mathematician John Horton Conway, who promoted its use, but essentially the same notation (using instead of ) can be found in an 1894 paper by Spanish mathematician . S = bc \\sin A = ac \\sin B = ab \\sin C , where \\begin{align} S &= 2 \\times \\text{reference triangle area}, \\\\[2pt] S_\\varphi &= S \\cot \\varphi. \\end{align} \\begin{align} S_A &= S \\cot A = bc \\cos A= \\frac {b^2+c^2-a^2} {2}, \\\\[2pt] S_B &= S \\cot B = ac \\cos B= \\frac {a^2+c^2-b^2} {2}, \\\\[2pt] S_C &= S \\cot C = ab \\cos C= \\frac {a^2+b^2-c^2} {2}, \\\\[2pt] S_\\omega &= S \\cot \\omega = \\frac {a^2+b^2+c^2} {2}, \\end{align} where is the Brocard angle. The law of cosines is used: a^2=b^2+c^2-2bc \\cos A. Third-, double-, and half-angle identities: \\begin{align} S_{\\frac {\\pi} {3}} &= S \\cot {\\frac {\\pi} {3}} = S \\frac {\\sqrt 3}{3} \\\\[2pt] S_{2\\varphi} &= \\frac {S_\\varphi^2 - S^2} {2S_\\varphi} \\\\[2pt] S_{ \\frac {\\varphi} {2}} &= S_\\varphi + \\sqrt {S_\\varphi^2 + S^2} \\end{align} for values of where , \\begin{align} S_{\\vartheta + \\varphi} &= \\frac {S_\\vartheta S_\\varphi - S^2} {S_\\vartheta + S_\\varphi}, \\\\[2pt] S_{\\vartheta - \\varphi} &= \\frac {S_\\vartheta S_\\varphi + S^2} {S_\\varphi - S_\\vartheta}. \\end{align} Furthermore the convention uses a shorthand notation for S_{\\vartheta}S_{\\varphi}=S_{\\vartheta\\varphi} , and S_{\\vartheta}S_{\\varphi}S_{\\psi}=S_{\\vartheta\\varphi\\psi} . \\begin{align} \\sin A &= \\frac {S} {bc} = \\frac {S} {\\sqrt {S_A^2 + S^2}} \\\\[2pt] \\cos A &= \\frac {S_A} {bc} = \\frac {S_A} {\\sqrt {S_A^2 + S^2}} \\\\[2pt] \\tan A &= \\frac {S} {S_A} \\\\[6pt] a^2 &= S_B + S_C \\\\[2pt] b^2 &= S_A + S_C \\\\[2pt] c^2 &= S_A + S_B \\end{align} \\begin{align} \\sum_\\text{cyclic} S_A &= S_A+S_B+S_C = S_\\omega \\\\[2pt] S^2 &= b^2c^2 - S_A^2 = a^2c^2 - S_B^2 = a^2b^2 - S_C^2 \\\\[6pt] S_{BC} &= S_BS_C = S^2 - a^2S_A \\\\[2pt] S_{AC} &= S_AS_C = S^2 - b^2S_B \\\\[2pt] S_{AB} &= S_AS_B = S^2 - c^2S_C \\\\[8pt] S_{ABC} &= S_AS_BS_C = S^2(S_\\omega-4R^2) \\\\[8pt] S_\\omega &= s^2-r^2-4rR \\end{align} where is the circumradius is the incenter abc = 2SR s = \\frac{a+b+c}{2} \\frac{S}{r} = a+b+c \\begin{align} \\sin A \\sin B \\sin C &= \\frac {S} {4R^2} \\\\[2pt] \\cos A \\cos B \\cos C &= \\frac {S_\\omega-4R^2} {4R^2} \\end{align} \\begin{align} \\sum_\\text{cyclic} \\sin A &= \\frac {S} {2Rr} = \\frac {s}{R} \\\\[2pt] \\sum_\\text{cyclic} \\cos A &= \\frac {r+R} {R} \\\\[2pt] \\sum_\\text{cyclic} \\tan A &= \\frac {S}{S_\\omega-4R^2}=\\tan A \\tan B \\tan C \\end{align} \\begin{array}{ll} \\displaystyle \\sum_\\text{cyclic} a^2S_A \\!\\! &=& a^2S_A + b^2S_B + c^2 S_C = 2S^2 \\\\[2pt] \\displaystyle \\sum_\\text{cyclic} a^4 &=& 2(S_\\omega^2-S^2) \\\\[2pt] \\displaystyle \\sum_\\text{cyclic} S_A^2 &=& S_\\omega^2 - 2S^2 \\\\[2pt] \\displaystyle \\sum_\\text{cyclic} S_{BC} &=& \\displaystyle \\sum_\\text{cyclic} S_BS_C = S^2 \\\\[2pt] \\displaystyle \\sum_\\text{cyclic} b^2c^2 &=& S_\\omega^2 + S^2 \\end{array} \\begin{align} P = p_a : p_b : p_c, \\\\ Q = q_a : q_b : q_c. \\end{align} Let \\begin{align} K_p &= ap_a + bp_b + cp_c, \\\\ K_q &= aq_a + bq_b + cq_c. \\end{align} Then is given by the formula: D^2 = \\sum_\\text{cyclic} a^2S_A\\left(\\frac {p_a}{K_p} - \\frac {q_a}{K_q}\\right)^2 For the circumcenter p_a = a S_A, and for the orthocenter q_a = \\frac{S_B S_C}{a}, \\begin{align} K_p &= \\sum_\\text{cyclic} a^2S_A = 2S^2, \\\\[2pt] K_q &= \\sum_\\text{cyclic} S_BS_C = S^2. \\end{align} Hence: \\begin{align} D^2 &= \\sum_\\text{cyclic} a^2S_A\\left(\\frac {aS_A} {2S^2} - \\frac {S_BS_C} {aS^2}\\right)^2 \\\\[2pt] &= \\frac {1} {4S^4} \\sum_\\text{cyclic} a^4S_A^3 - \\frac {S_AS_BS_C} {S^4} \\sum_\\text{cyclic} a^2S_A + \\frac {S_AS_BS_C} {S^4} \\sum_\\text{cyclic} S_BS_C \\\\[2pt] &= \\frac {1} {4S^4} \\sum_\\text{cyclic} a^2S_A^2(S^2-S_BS_C) - 2(S_\\omega-4R^2) + (S_\\omega-4R^2) \\\\[2pt] &= \\frac {1} {4S^2} \\sum_\\text{cyclic} a^2S_A^2 - \\frac {S_AS_BS_C} {S^4} \\sum_\\text{cyclic} a^2S_A - (S_\\omega-4R^2) \\\\[2pt] &= \\frac {1} {4S^2} \\sum_\\text{cyclic} a^2(b^2c^2-S^2) - \\frac {1} {2}(S_\\omega-4R^2) -(S_\\omega-4R^2) \\\\[2pt] &= \\frac {3a^2b^2c^2} {4S^2} - \\frac {1} {4} \\sum_\\text{cyclic} a^2 - \\frac {3} {2}(S_\\omega-4R^2) \\\\[2pt] &= 3R^2- \\frac {1} {2} S_\\omega - \\frac {3} {2} S_\\omega + 6R^2 \\\\[10pt] &= 9R^2- 2S_\\omega. \\end{align} Thus, |OH| = \\sqrt{9R^2- 2S_\\omega}.",
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    "title": "Conway triangle notation",
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        "family": "Anticloud FZ LLE",
        "literal": "Anticloud FZ LLE"
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    "container-title": "encyclopegia.org",
    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/conway-triangle-notation",
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      "date-parts": [
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          15
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