Plate notation - Revision history

en>Quebec99: fixed reference

2014-03-29T17:49:50Z

fixed reference

← Older revision		Revision as of 19:49, 29 March 2014
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]~~]

en>Jonesey95: Fixing "CS1 errors: ISSN" error in citation.

2013-11-27T05:28:07Z

Fixing "CS1 errors: ISSN" error in citation.

← Older revision		Revision as of 07:28, 27 November 2013
Line 1:		Line 1:
	~~Today~~, ~~there are several other types~~ of ~~web development and blogging software available~~ to ~~design and host your website blogs online and that too in minutes~~, ~~if not hours. You can either install Word - Press yourself or use free services offered on the web today. One really cool features about this amazing~~ and ~~free wp plugin is that the code it generates is completely portable. If you~~'~~re using Wordpress~~ and ~~want to make your blog a "dofollow" blog~~, ~~meaning that links from your blog pass on the benefits of Google pagerank~~, ~~you can install one of the many dofollow plugins available. If you are happy with your new look then click "Activate~~ '~~New Theme~~'~~" in~~ the ~~top right corner. <br><br>~~		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:

	~~As you know today Word~~ - ~~Press has turn out to be~~ a ~~tremendously popular open source publishing and blogging display place. You may either choose to link only to the top~~-~~level category pages or~~ the ~~ones that contain information about your products and services~~. ~~You are able to set them within your theme options and so they aid the search engine to get a suitable title and description~~ for ~~the pages that get indexed by Google.~~ ~~In case you have any questions concerning where by and also how you can use [http~~://~~pajuegos.com~~/~~profile~~/~~dachild wordpress backup]~~, it is ~~possible to call us on our own website. These four plugins will make this effort easier~~ and the ~~sites run effectively as well as make other widgets added to~~ a ~~site easier to configure. This can be done by using~~ a ~~popular layout format and your unique Word~~ - ~~Press design can be achieved in other elements of the blog.~~ <br><br>~~The entrepreneurs can easily captivate their readers by using these versatile themes. After sending these details~~, ~~your Word~~ - ~~Press blog will~~ be ~~setup within~~ a ~~few days~~. This platform can be customizedaccording to the requirements of the business. These frequent updates have created menace in the task of optimization. Search engine optimization pleasant picture and ~~solution links suggest you will have~~ a ~~much better adjust at gaining considerable natural site visitors.~~ <br><br>~~Whether your Word - Press themes~~ is ~~premium or not, but nowadays every theme is designed with widget-ready. Cameras with~~ a ~~pentaprism~~ (~~as in comparison to pentamirror~~) ~~ensure that little mild~~ is ~~lost before it strikes your eye~~, ~~however these often increase~~ the ~~cost of~~ the ~~digital camera considerably. However, you may not be able to find~~ a ~~theme that is in sync with your business. Fast Content Update - It~~'~~s easy to edit or add posts with free Wordpress websites. The Pakistani culture is in demand of a main surgical treatment.~~ <br><br>~~More it extends numerous opportunities where your firm is at comfort~~ and ~~rest assured of no risks~~ & ~~errors. Being~~ a ~~Plugin Developer~~, ~~it is important for you to know that development of Word~~ - ~~Press driven website should be done only when you enable debugging. It's not~~ a ~~secret that~~ a ~~lion share of activity on the internet is takes place on the Facebook. Thus, Word~~ - ~~Press is~~ a ~~good alternative if you are looking for free blogging software~~. ~~Definitely when you wake up from the slumber~~, ~~you can be sure that you will be lagging behind and getting on track would be a tall order~~.		:<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]]

en>Benwing: /* Extensions */ fix links

2012-04-02T09:59:41Z

Extensions: fix links

New page

Today, there are several other types of web development and blogging software available to design and host your website blogs online and that too in minutes, if not hours. You can either install Word - Press yourself or use free services offered on the web today. One really cool features about this amazing and free wp plugin is that the code it generates is completely portable. If you're using Wordpress and want to make your blog a "dofollow" blog, meaning that links from your blog pass on the benefits of Google pagerank, you can install one of the many dofollow plugins available. If you are happy with your new look then click "Activate 'New Theme'" in the top right corner. 

As you know today Word - Press has turn out to be a tremendously popular open source publishing and blogging display place. You may either choose to link only to the top-level category pages or the ones that contain information about your products and services. You are able to set them within your theme options and so they aid the search engine to get a suitable title and description for the pages that get indexed by Google. In case you have any questions concerning where by and also how you can use [http://pajuegos.com/profile/dachild wordpress backup], it is possible to call us on our own website. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a site easier to configure. This can be done by using a popular layout format and your unique Word - Press design can be achieved in other elements of the blog. The entrepreneurs can easily captivate their readers by using these versatile themes. After sending these details, your Word - Press blog will be setup within a few days. This platform can be customizedaccording to the requirements of the business. These frequent updates have created menace in the task of optimization. Search engine optimization pleasant picture and solution links suggest you will have a much better adjust at gaining considerable natural site visitors. Whether your Word - Press themes is premium or not, but nowadays every theme is designed with widget-ready. Cameras with a pentaprism (as in comparison to pentamirror) ensure that little mild is lost before it strikes your eye, however these often increase the cost of the digital camera considerably. However, you may not be able to find a theme that is in sync with your business. Fast Content Update - It's easy to edit or add posts with free Wordpress websites. The Pakistani culture is in demand of a main surgical treatment. More it extends numerous opportunities where your firm is at comfort and rest assured of no risks & errors. Being a Plugin Developer, it is important for you to know that development of Word - Press driven website should be done only when you enable debugging. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. Thus, Word - Press is a good alternative if you are looking for free blogging software. Definitely when you wake up from the slumber, you can be sure that you will be lagging behind and getting on track would be a tall order.