Photon diffusion equation - Revision history

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Reverted edits by 117.213.8.191 (talk) to last version by Addbot

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Work with a cooking apply, lіke Pam, as oppoѕed to buttеr. Tension and rinse making use~~ of ~~hot water when browning meat~~. ~~This [~~http://www.~~gortzhausgallery~~.~~com~~/~~Default~~.~~aspx?tabid~~=~~92&userId~~=~~52178 where can i buy vigrx plus in stores] get rid οf a lot of~~ the ~~unwanted fat~~.<br><br>It cаn bе an easy task to feel you aгe consuming effectively when you are not. Small miscalϲulations can cause being overweight. This short article offered you some good recommendations. Use each of the stratеgies that you may have figured out for fantastic health.		[[Image:Archimedes' Quadruplets.svg\|thumb\|right\|400px\|Each of the Archimedes' quadruplets (green) have equal area to each other and to Archimedes' twin circles]]
			In [[geometry]], '''Archimedes' quadruplets''' are four [[Congruence (geometry)\|congruent]] [[circle]]s associated with an [[arbelos]]. Introduced by Frank Power in the summer of 1998, each have the same [[area]] as [[Archimedes' twin circles]], making them [[Archimedean circle]]s.<ref>
			{{ citation
			\| last=Power
			\| first=Frank
			\| title=Forum Geometricorum
			\| volume=5
			\| chapter=Some More Archimedean Circles in the Arbelos
			\| date=2005
			\| publication-date=2005-11-02
			\| editor-last=Yiu
			\| editor-first=Paul
			\| pages=133–134
			\| issn=1534-1178
			\| url=http://forumgeom.fau.edu/FG2005volume5/FG200517.ps
			\| accessdate=2008-04-13
			}}</ref>

			==Construction==

			An arbelos is formed from three collinear points ''A'', ''B'', and ''C'', by the three [[semicircle]]s with [[diameter]]s ''AB'', ''AC'', and ''BC''. Let the two smaller circles have [[radii]] ''r''<sub>1</sub> and ''r''<sub>2</sub>, from which it follows that the larger semicircle has radius ''r'' = ''r''<sub>1</sub>+''r''<sub>2</sub>. Let the points ''D'' and ''E'' be the [[centre (geometry)\|center]] and [[midpoint]], respectively, of the semicircle with the radius ''r''<sub>1</sub>. Let ''H'' be the midpoint of line ''AC''. Then two of the four quadruplet circles are tangent to line ''HE'' at the point ''E'', and are also tangent to the outer semicircle. The other two quadruplet circles are formed in a symmetric way from the semicircle with radius ''r''<sub>2</sub>.

			==Proof of congruency==

			According to Proposition 5 of [[Archimedes]]' ''[[Book of Lemmas]]'', the common [[radius]] of Archimedes' twin circles is:

			:<math>\frac{r_1\cdot r_2}{r}.</math>

			By the [[Pythagorean theorem]]:

			:<math>\left(HE\right)^2=\left(r_1\right)^2+\left(r_2\right)^2.</math>

			Then, create two circles with centers ''J<sub>i</sub>'' [[perpendicular]] to ''HE'', [[tangent]] to the large semicircle at point ''L<sub>i</sub>'', tangent to point ''E'', and with equal radii ''x''. Using the [[Pythagorean theorem]]:

			:<math>\left(HJ_i\right)^2=\left(HE\right)^2+x^2=\left(r_1\right)^2+\left(r_2\right)^2+x^2</math>

			Also:

			:<math>HJ_i=HL_i-x=r-x=r_1+r_2-x~</math>

			Combinding these gives:

			:<math>\left(r_1\right)^2+\left(r_2\right)^2+x^2=\left(r_1+r_2-x\right)^2</math>

			Expanding, collecting to one side, and factoring:

			:<math>2r_1r_2-2x\left(r_1+r_2\right)=0</math>

			Solving for ''x'':

			:<math>x=\frac{r_1\cdot r_2}{r_1+r_2}=\frac{r_1\cdot r_2}{r}</math>

			Proving that each of the Archimedes' quadruplets' areas is equal to each of Archimedes' twin circles' areas.<ref>
			{{citeweb
			\| last=Bogomolny
			\| first=Alexander
			\| title=Archimedes' Quadruplets
			\| url=http://www.cut-the-knot.org/Curriculum/Geometry/ArchimedesQuadruplets.shtml
			\| accessdate=2008-04-13
			\| archiveurl= http://web.archive.org/web/20080512195739/http://www.cut-the-knot.org/Curriculum/Geometry/ArchimedesQuadruplets.shtml\| archivedate= 12 May 2008 <!--DASHBot-->\| deadurl= no}}</ref>

			==References==
			{{reflist}}

			[[Category:Arbelos]]

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