GetWiki
cevian
ARTICLE SUBJECTS
being →
database →
ethics →
fiction →
history →
internet →
language →
linux →
logic →
method →
news →
policy →
purpose →
religion →
science →
software →
truth →
unix →
wiki →
ARTICLE TYPES
essay →
feed →
help →
system →
wiki →
ARTICLE ORIGINS
critical →
forked →
imported →
original →
cevian
please note:
- the content below is remote from Wikipedia
- it has been imported raw for GetWiki
{{short description|Line intersecting both a vertex and opposite edge of a triangle}}In geometry, a cevian is a line segment which joins a vertex of a triangle to a point on the opposite side of the triangle.BOOK, Coxeter, H. S. M., Harold Scott MacDonald Coxeter, Greitzer, S. L., Samuel L. Greitzer, 1967, Geometry Revisited,weblink limited, Washington, DC, Mathematical Association of America, 0-883-85619-0, 4, Some authors exclude the other two sides of the triangle, see {{harvtxt|Eves|1963|loc=p.77}} Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovanni Ceva, who proved a well-known theorem about cevians which also bears his name.JOURNAL, Lightner, James E., 1975, A new look at the 'centers' of a triangle, The Mathematics Teacher, 68, 7, 612â615, 27960289, - the content below is remote from Wikipedia
- it has been imported raw for GetWiki
Length
right|thumb|A triangle with a cevian of length {{mvar|d}}Stewart's theorem
The length of a cevian can be determined by Stewart's theorem: in the diagram, the cevian length {{mvar|d}} is given by the formula
,b^2m + c^2n = a(d^2 + mn).
Less commonly, this is also represented (with some rearrangement) by the following mnemonic:
underset{text{A }mantext{ and his }dad}{man + dad} = !!!!!! underset{text{put a }bombtext{ in the }sink.}{bmb + cnc}WEB,weblink's_Theorem, Art of Problem Solving, artofproblemsolving.com, 2018-10-22,
Median
If the cevian happens to be a median (thus bisecting a side), its length can be determined from the formula
,m(b^2 + c^2) = a(d^2 + m^2)
or
,2(b^2 + c^2) = 4d^2 + a^2
since
,a = 2m.
Hence in this case
d= fracsqrt{2 b^2 + 2 c^2 - a^2}2 .
Angle bisector
If the cevian happens to be an angle bisector, its length obeys the formulas
,(b + c)^2 = a^2 left( frac{d^2}{mn} + 1 right),
andJohnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929), p. 70.
d^2+mn = bc
and
d= frac{2 sqrt{bcs(s-a)}}{b+c}
where the semiperimeter s = tfrac{a+b+c}{2}.The side of length {{math|a}} is divided in the proportion {{math|b : c}}.Altitude
If the cevian happens to be an altitude and thus perpendicular to a side, its length obeys the formulas
,d^2 = b^2 - n^2 = c^2 - m^2
and
d=frac{2sqrt{s(s-a)(s-b)(s-c)}}{a},
where the semiperimeter s = tfrac{a+b+c}{2}.Ratio properties
(File:Ceva's theorem 1.svg|thumb|right|Three cevians passing through a common point)There are various properties of the ratios of lengths formed by three cevians all passing through the same arbitrary interior point:Alfred S. Posamentier and Charles T. Salkind, Challenging Problems in Geometry, Dover Publishing Co., second revised edition, 1996.{{rp|177-188}} Referring to the diagram at right,
begin{align}
& frac{overline{AF}}{overline{FB}} cdot frac{overline{BD}}{overline{DC}} cdot frac{overline{CE}}{overline{EA}} = 1 & & frac{overline{AO}}{overline{OD}} = frac{overline{AE}}{overline{EC}} + frac{overline{AF}}{overline{FB}}; & & frac{overline{OD}}{overline{AD}} + frac{overline{OE}}{overline{BE}} + frac{overline{OF}}{overline{CF}} = 1; & & frac{overline{AO}}{overline{AD}} + frac{overline{BO}}{overline{BE}} + frac{overline{CO}}{overline{CF}} = 2.end{align}The first property is known as Ceva's theorem. The last two properties are equivalent because summing the two equations gives the identity {{math|1=1 + 1 + 1 = 3}}.Splitter
A splitter of a triangle is a cevian that bisects the perimeter. The three splitters concur at the Nagel point of the triangle.Area bisectors
Three of the area bisectors of a triangle are its medians, which connect the vertices to the opposite side midpoints. Thus a uniform-density triangle would in principle balance on a razor supporting any of the medians.Angle trisectors
If from each vertex of a triangle two cevians are drawn so as to trisect the angle (divide it into three equal angles), then the six cevians intersect in pairs to form an equilateral triangle, called the Morley triangle.Area of inner triangle formed by cevians
Routh's theorem determines the ratio of the area of a given triangle to that of a triangle formed by the pairwise intersections of three cevians, one from each vertex.See also
Notes
{{reflist}}References
- {{citation|first=Howard|last=Eves|title=A Survey of Geometry (Vol. One)|publisher=Allyn and Bacon|year=1963}}
- Ross Honsberger (1995). Episodes in Nineteenth and Twentieth Century Euclidean Geometry, pages 13 and 137. Mathematical Association of America.
- Vladimir Karapetoff (1929). "Some properties of correlative vertex lines in a plane triangle." American Mathematical Monthly 36: 476â479.
- Indika Shameera Amarasinghe (2011). âA New Theorem on any Right-angled Cevian Triangle.â Journal of the World Federation of National Mathematics Competitions, Vol 24 (02), pp. 29â37.
- content above as imported from Wikipedia
- "cevian" does not exist on GetWiki (yet)
- time: 5:31pm EDT - Wed, May 01 2024
- "cevian" does not exist on GetWiki (yet)
- time: 5:31pm EDT - Wed, May 01 2024
[ this remote article is provided by Wikipedia ]
LATEST EDITS [ see all ]
GETWIKI 23 MAY 2022
The Illusion of Choice
Culture
Culture
GETWIKI 09 JUL 2019
Eastern Philosophy
History of Philosophy
History of Philosophy
GETWIKI 09 MAY 2016
GetMeta:About
GetWiki
GetWiki
GETWIKI 18 OCT 2015
M.R.M. Parrott
Biographies
Biographies
GETWIKI 20 AUG 2014
GetMeta:News
GetWiki
GetWiki
© 2024 M.R.M. PARROTT | ALL RIGHTS RESERVED