|
|
| Line 1: |
Line 1: |
| In [[geometry]], the '''Conway triangle notation''', named after [[John Horton Conway]], allows [[trigonometric functions]] of a [[triangle]] to be managed algebraically. Given a reference triangle whose sides are ''a'', ''b'' and ''c'' and whose corresponding internal [[angle]]s are ''A'', ''B'', and ''C'' then the Conway triangle notation is simply represented as follows:
| | I am Zandra from Almhult. I love to play Clarinet. Other hobbies are Art collecting.<br><br>Here is my site :: [https://deathcontrol.de/index/users/view/id/49661 wordpress dropbox backup] |
| | |
| :<math> S = bc \sin A = ac \sin B = ab \sin C \,</math>
| |
| | |
| where ''S'' = 2 × area of reference triangle and
| |
|
| |
| :<math> S_\varphi = S \cot \varphi . \,</math>
| |
| | |
| in particular
| |
| | |
| :<math> S_A = S \cot A = bc \cos A= \frac {b^2+c^2-a^2} {2}\,</math>
| |
| | |
| :<math> S_B = S \cot B = ac \cos B= \frac {a^2+c^2-b^2} {2}\,</math>
| |
| | |
| :<math> S_C = S \cot C = ab \cos C= \frac {a^2+b^2-c^2} {2}\,</math>
| |
| | |
| :<math> S_\omega = S \cot \omega = \frac {a^2+b^2+c^2} {2}\,</math> where <math> \omega \,</math> is the [[Brocard angle]].
| |
| | |
| :<math> S_{\frac {\pi} {3}} = S \cot {\frac {\pi} {3}} = S \frac {\sqrt 3}{3} \,</math> | |
| | |
| :<math> S_{2\varphi} = \frac {S_\varphi^2 - S^2} {2S_\varphi} \quad\quad S_{ \frac {\varphi} {2}} = S_\varphi + \sqrt {S_\varphi^2 + S^2} \,</math> for values of <math> \varphi </math> where <math> 0 < \varphi < \pi \, </math> | |
| | |
| :<math> S_{\vartheta + \varphi} = \frac {S_\vartheta S_\varphi - S^2} {S_\vartheta + S_\varphi} \quad\quad S_{\vartheta - \varphi} = \frac {S_\vartheta S_\varphi + S^2} {S_\varphi - S_\vartheta} \, </math>
| |
| | |
| Hence:
| |
| | |
| :<math> \sin A = \frac {S} {bc} = \frac {S} {\sqrt {S_A^2 + S^2}} \quad\quad \cos A = \frac {S_A} {bc} = \frac {S_A} {\sqrt {S_A^2 + S^2}} \quad\quad \tan A = \frac {S} {S_A} \, </math>
| |
| | |
| Some important identities:
| |
| | |
| :<math> \sum_\text{cyclic} S_A = S_A+S_B+S_C = S_\omega \, </math>
| |
| | |
| :<math> S^2 = b^2c^2 - S_A^2 = a^2c^2 - S_B^2 = a^2b^2 - S_C^2 \, </math>
| |
| | |
| :<math> S_BS_C = S^2 - a^2S_A\quad\quad S_AS_C = S^2 - b^2S_B\quad\quad S_AS_B = S^2 - c^2S_C \, </math>
| |
| | |
| :<math> S_AS_BS_C = S^2(S_\omega-4R^2)\quad\quad S_\omega=s^2-r^2-4rR \, </math>
| |
| | |
| where ''R'' is the [[circumcenter|circumradius]] and ''abc'' = 2''SR'' and where ''r'' is the [[incenter]], <math> s= \frac{a+b+c}{2} \, </math> and <math> a+b+c = \frac {S} {r} \, </math>
| |
| | |
| Some useful trigonometric conversions:
| |
| | |
| :<math> \sin A \sin B \sin C = \frac {S} {4R^2} \quad\quad \cos A \cos B \cos C = \frac {S_\omega-4R^2} {4R^2} </math> | |
| :<math> \sum_\text{cyclic} \sin A = \frac {S} {2Rr} = \frac {s}{R} \quad\quad \sum_\text{cyclic} \cos A = \frac {r+R} {R} \quad\quad \sum_\text{cyclic} \tan A = \frac {S}{S_\omega-4R^2}=\tan A \tan B \tan C \, </math>
| |
| | |
| | |
| Some useful formulas:
| |
| | |
| :<math> \sum_\text{cyclic} a^2S_A = a^2S_A + b^2S_B + c^2 S_C = 2S^2 \quad\quad \sum_\text{cyclic} a^4 = 2(S_\omega^2-S^2) \, </math>
| |
| | |
| :<math> \sum_\text{cyclic} S_A^2 = S_\omega^2 - 2S^2 \quad\quad \sum_\text{cyclic} S_BS_C = S^2 \quad\quad \sum_\text{cyclic} b^2c^2 = S_\omega^2 + S^2 \, </math>
| |
| | |
| Some examples using Conway triangle notation:
| |
| | |
| Let ''D'' be the distance between two points P and Q whose [[trilinear coordinates]] are ''p''<sub>''a''</sub> : ''p''<sub>''b''</sub> : ''p''<sub>''c''</sub> and ''q''<sub>''a''</sub> : ''q''<sub>''b''</sub> : ''q''<sub>''c''</sub>. Let ''K''<sub>''p''</sub> = ''ap''<sub>''a''</sub> + ''bp''<sub>''b''</sub> + ''cp''<sub>''c''</sub> and let ''K''<sub>''q''</sub> = ''aq''<sub>''a''</sub> + ''bq''<sub>''b''</sub> + ''cq''<sub>''c''</sub>. Then ''D'' is given by the formula:
| |
| | |
| :<math> D^2= \sum_\text{cyclic} a^2S_A\left(\frac {p_a}{K_p} - \frac {q_a}{K_q}\right)^2 \, </math>
| |
| | |
| Using this formula it is possible to determine OH, the distance between the circumcenter and the [[orthocenter]] as follows:
| |
| | |
| For the circumcenter ''p''<sub>''a''</sub> = ''aS''<sub>''A''</sub> and for the orthocenter ''q''<sub>''a''</sub> = ''S''<sub>''B''</sub>''S''<sub>''C''</sub>/''a''
| |
| :<math> K_p= \sum_\text{cyclic} a^2S_A = 2S^2 \quad\quad K_q= \sum_\text{cyclic} S_BS_C = S^2 \,</math>
| |
| | |
| Hence:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| D^2 & {} = \sum_\text{cyclic} a^2S_A\left(\frac {aS_A} {2S^2} - \frac {S_BS_C} {aS^2}\right)^2 \\
| |
| & {} = \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 \\
| |
| & {} = \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) \\
| |
| & {} = \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) \\
| |
| & {} = \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) \\
| |
| & {} = \frac {3a^2b^2c^2} {4S^2} - \frac {1} {4} \sum_\text{cyclic} a^2 - \frac {3} {2}(S_\omega-4R^2) \\
| |
| & {} = 3R^2- \frac {1} {2} S_\omega - \frac {3} {2} S_\omega + 6R^2 \\
| |
| & {} = 9R^2- 2S_\omega.
| |
| \end{align}
| |
| </math>
| |
| | |
| This gives:
| |
| | |
| :<math> OH = \sqrt{9R^2- 2S_\omega \,}.</math>
| |
| | |
| ==References==
| |
| * {{mathworld|urlname=ConwayTriangleNotation|title=Conway Triangle Notation}}
| |
| | |
| [[Category:Triangle geometry]]
| |
| [[Category:Trigonometry]]
| |
I am Zandra from Almhult. I love to play Clarinet. Other hobbies are Art collecting.
Here is my site :: wordpress dropbox backup