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	Symmedian - Wikipedia, the free encyclopedia
		The symmedians intersect in the Lemoine point L. The symmedian point of a triangle with sides a, b and c has homogeneous trilinear coordinates [a : b : c].
		The symmedian point L can also be constructed differently: the three lines joining the midpoint of a side to the midpoint of the altitude on that side intersect in L.
		The symmedian point is the isogonal conjugate of the triangle's centroid.
	en.wikipedia.org /wiki/Symmedian (214 words)

	PlanetMath: symmedian
		On any triangle, the three lines obtained by reflecting the medians in the (internal) angle bisectors are called the symmedians of the triangle.
		It can be stated as symmedians are isogonal conjugates of medians.
		This is version 2 of symmedian, born on 2002-01-08, modified 2002-05-16.
	planetmath.org /encyclopedia/Symmedian.html (86 words)

	Symmedian and 2 Antiparallels
		In a triangle ABC the antiparallels to sides AB and AC that meet on the symmedian from C have equal lengths.
		Let CS be the symmedian and MR and LN the two antiparallels in question that meet in point T on CS.
		By transitivity, the three antiparallels through the symmedian point all have equal lengths.
	www.cut-the-knot.org /Curriculum/Geometry/Sym2Antiparallel.shtml (117 words)

	Symmedian -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		The symmedian point L can also be constructed differently: the three lines joining the midpoint of a side to the midpoint of the (Elevation especially above sea level or above the earth's surface) altitude on that side intersect in L.
		The symmedian point of a (A triangle with one right angle) right triangle is therefore the midpoint of the altitude on the hypotenuse.
		The (Click link for more info and facts about Gergonne point) Gergonne point of a triangle is the same as the symmedian point of the triangle's (Click link for more info and facts about contact triangle) contact triangle.
	www.absoluteastronomy.com /encyclopedia/s/sy/symmedian.htm (240 words)

	Symmedian (Site not responding. Last check: 2007-10-21)
		The symmedians intersect in the Lemoinepoint L. The symmedian point of a triangle with sides a, b and c has homogeneous trilinear coordinates [a : b : c].
		The symmedian point L can also be constructed differently: the three lines joining the midpoint of a side to themidpoint of the altitude on that side intersect inL.
		The symmedian point of a right triangle is therefore themidpoint of the altitude on the hypotenuse.
	www.therfcc.org /symmedian-214600.html (185 words)

	Symmedian and Antiparallel
		A symmedian drawn from a vertex of a triangle divides the antiparallels to the opposite side in half.
		It follows that CS (CT extended) is the symmedian from C. This result is often presented in a different form: the locus of the midpoints of the antiparallels to a side of a triangle is the summedian through the opposite vertex.
		In a right-angled triangle the symmedian point coincides with the midpoint of the altitude to the hypotenuse.
	www.cut-the-knot.org /Curriculum/Geometry/SymAntiparallel.shtml (215 words)

	Symmedian (Site not responding. Last check: 2007-10-21)
		The symmedians intersect in Lemoine point L. The symmedian point of a triangle with a b and c has homogeneous trilinear coordinates [ a : b : c ].
		The symmedian point L can also be constructed differently: the lines joining the midpoint of a side the midpoint of the altitude on that side intersect in L.
		The symmedian point of a right triangle is therefore the midpoint of the on the hypotenuse.
	www.freeglossary.com /Symmedian (503 words)

	Unusual Properties of Triangles
		A symmedian of a triangle from vertex A is obtained by reflecting the median from A in the bisector of the angle A.
		Among other results on symmedians in Lemoine's 1873 paper he showed that the symmedian from the vertex A cuts the side BC of the triangle in the ratio of the squares of the sides AC and AB.
		These were 7.81 inches for the left side, 5.09 inches for the right side, 4.91 inches from the left vertex to the crossing of the symmedian line with the base, and 2.09 inches from the crossing of the symmedian line on the base to the right vertex.
	www.egyptorigins.com /unusualproperties.htm (1089 words)

	Antiparallels (Site not responding. Last check: 2007-10-21)
		The middles (M) of the segments JH move on a line CK, which is the [symmedian] of the triangle w.r.
		Hence the Symmedian point of ABC (which is the intersection point of the three symmedians (look at Symmedian.html) is the Gergonne point of its tangential triangle.
		Moving M to the position, such that AB = JH (make F and G coincident), we see that the symmedian is the symmetric of the median from C, with respect to the bisector of C. Look at the file Symmedian_0.html for a more detailed discussion of the symmedian line from a vertex and its properties.
	www.math.uoc.gr /~pamfilos/eGallery/problems/Antiparallels.html (184 words)

	PlanetMath: fundamental theorem on isogonal lines
		An applications of this theorem proves the existence of Lemoine point (for it is the intersection point of the symmedians):
		This theorem is a direct consequence of Ceva's theorem (trigonometric version).
		This is version 1 of fundamental theorem on isogonal lines, born on 2002-09-02.
	planetmath.org /encyclopedia/FundamentalTheoremOnIsogonalLines.html (83 words)

	Triangle - Wikipedia, the free encyclopedia
		The center of the incircle is not in general located on Euler's line.
		If one reflects a median at the angle bisector that passes through the same vertex, one obtains a symmedian.
		The three symmedians intersect in a single point, the symmedian point of the triangle.
	www.wikipedia.org /wiki/Triangle (2072 words)

	Darij Grinberg
		Let the symmedian AL of triangle ABC meet the side BC at a point D' and the circumcircle of triangle ABC at a point X' (different from A).
		Two proofs of the fact that the symmedian point is the centroid of its pedal triangle are presented: The first one uses the tangential triangle; the second one is a rather standard proof.
		A new proof is given for the fact that the symmedian point of a triangle is the symmedian point of its circumcevian triangle.
	de.geocities.com /darij_grinberg (3178 words)

	Encyclopedia: Symmedian (Site not responding. Last check: 2007-10-21)
		For alternate meanings, such as the musical instrument, see triangle (disambiguation).
		Symmedians (red) of a triangle intersecting in the symmedian point L. File downloaded from PlanetMath at http://planetmath.
		The isogonal conjugate of a point P with respect to a triangle ABC is constructed by reflecting the lines PA, PB, and PC about the angle bisectors of A, B, and C. These three reflected lines concur at the isogonal conjugate of P. Categories: Geometry stubs...
	www.nationmaster.com /encyclopedia/Symmedian (475 words)

	PlanetMath: Lemoine point
		The Lemoine point of a triangle, is the intersection point of its three symmedians.
		See Also: triangle, symmedian, Lemoine circle, incircle, centroid, incenter, Gergonne point, isogonal, isogonal conjugate, fundamental theorem on isogonal lines
		Cross-references: incentre, medians, lines, BLUE, Gergonne triangle, Gergonne point, centroid, isogonal conjugate, symmedians, intersection, triangle
	www.planetmath.org /encyclopedia/LemoinePoint.html (123 words)

	Symmedian (Site not responding. Last check: 2007-10-21)
		The symmedian at C, of a triangle ABC, is defined as the symmetric line of the median from C, with respect to the bisector of C. The three symmedians of ABC pass through the [Symmedian] point K of the triangle (look at the file: Antiparallels.html).
		Look at the file Symmedian_0.html for a more detailed discussion of the symmedian line from a vertex and its properties.
		With this subject is related the picture of three Apollonian circles, in the file: Isodynamic.html.
	www.math.uoc.gr /~pamfilos/eGallery/problems/Symmedian.html (146 words)

	EMILE LEMOINE (Site not responding. Last check: 2007-10-21)
		In 1883, M. d'Ocagne introduced the term symmedian for a line obtained as the reflection of a median of a triangle about the corresponding angle bisector (i.e., Lemoine's "antiparallel median".
		As the LHuilier-Lemoine-Grebe point is the point of concurrence of the three symmedians of a triangle, the geometer Robert Tucker (as in Tucker circles) introduced yet another name for the point: symmedian point (link below).
		Fifteen years after the momentous 1873 paper, Lemoine presented another distinctive development to the Association Française pour l'Avancement des Sciences [meeting in Oran, Algeria].
	faculty.evansville.edu /ck6/bstud/lemoine.html (556 words)

	Symmedian - TheBestLinks.com - Symmedian point, Geometry, Right triangle, Straight line, ... (Site not responding. Last check: 2007-10-21)
		Symmedian - TheBestLinks.com - Symmedian point, Geometry, Right triangle, Straight line,...
		Symmedian point, Symmedian, Geometry, Right triangle, Straight line, Altitude...
		You can add this article to your own "watchlist" and receive e-mail notification about all changes in this page.
	www.thebestlinks.com /Symmedian_point.html (247 words)

	symmedian (Site not responding. Last check: 2007-10-21)
		A' is on the angle bisector of angle A. M is the midpoint of BC, so AM is the median.
		The blue line is the Symmedian (the isogonal conjugate of the median).
		B' and C' are on lines AB and AC, and M' is the intersection of the symmedian with line B'C'.
	www.uwrf.edu /~ll63/sketch/symmedian.html (133 words)

	Lemoine (Site not responding. Last check: 2007-10-21)
		He founded a new study of properties of a triangle in a paper of 1873 where he studied the point of intersection of the symmedians of a triangle.
		Among other results on symmedians in Lemoine's 1873 paper is the result that the symmedian from the vertex A cuts the side BC of the triangle in the ratio of the squares of the sides AC and AB.
		All I [EFR] can add is that I agree with the mathematicians of the time who preferred a construction with a large number of easily understood steps to a shorter one with sophisticated, rather obscure, steps.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Lemoine.html (629 words)

	symmedian - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "symmedian" is defined.
		Symmedian : A Glossary of Mathematical Terms [home, info]
		Symmedian : Eric Weisstein's World of Mathematics [home, info]
	onelook.com /?w=symmedian (86 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-21)
		The three lines Ga, Gb, Gc meet in the symmedian point.
		Date: 11/18/2002 at 09:08:06 From: Doctor Floor Subject: Re: Symmedian point Hi, Lina, Thanks for your question.
		To prove that the three lines Ga, Gb, and Gc are concurrent, we use the trigonometric form of Ceva's theorem.
	mathforum.org /library/drmath/view/61701.html (150 words)

	MathLinks Math Forum :: View topic - Symmedian and circumcircles
		Since < BAM = < DAC, the line AM is the reflection of the line AD in the angle bisector of the angle CAB.
		But the line AD is a median of triangle ABC; hence, the line AM is a symmedian of triangle ABC.
		Since a symmedian from a vertex of a triangle divides the opposite side in the ratio of the squares of the two adjacent sides, we have
	www.mathlinks.ro /Forum/topic-20166.html (509 words)

	Mathh 114 Lab 1 (Site not responding. Last check: 2007-10-21)
		With Geometer's Sketchpad, produce a script that accepts three points and which returns the nine-point circle, the three sets of three significant points (each triplet displayed in a special color), the Euler segment, and the four significant points on the Euler segment.
		Produce a script that accepts three points and which returns the symmedian point.
		Produce a script which accepts three points, and which returns the Brocard circle, the diameter from the circumcenter and symmedian point, and the two Brocard points.
	www.uwec.edu /smithaj/fall98/330/labs/lab02/lab.html (134 words)

	FG200102index (Site not responding. Last check: 2007-10-21)
		Again the six vertices are concyclic, and the process may be repeated indefinitely to form an infinite family of circles (Tucker [3]).
		This family is a coaxaloid system, and its locus of centers is the Brocard axis of the triangle, passing through the circumcenter and the symmedian point.
		Third [2] extended this idea by relaxing the conditions for the directions of the sides of the hexagon, thus finding infinitely many coaxaloid systems of circles.
	forumgeom.fau.edu /FG2002volume2/FG200210index.html (207 words)

	Symmedian - Encyclopedia Glossary Meaning Explanation Symmedian (Site not responding. Last check: 2007-10-21)
		Here you will find more informations about Symmedian.
		If you find this encyclopedia or its sister projects useful,
		The symmedians intersect in the Lemoine point L. The symmedian point of a triangle with sides a, b and c has homogeneous trilinear coordinates a" target="_blank" class="exlink">: b : c.
	www.encyclopedia-glossary.com /en/Symmedian.html (220 words)

	ENCYCLOPEDIA OF TRIANGLE CENTERS
		X(6) is the point of concurrence of the symmedians (reflections of medians in corresponding angle bisectors); the point (x, y, z), given here in actual trilinear distances, that minimizes x
		If you have The Geometer's Sketchpad, you can view Symmedian point.
		X(9) is the symmedian point of the excentral triangle.
	faculty.evansville.edu /ck6/encyclopedia/ETC.html (10314 words)

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