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Topic: Isogonal conjugate

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	GLOSSARY, a support page for ENCYCLOPEDIA TRIANGLE CENTERS
		Details on gimel conjugates, as well as aleph, beth, daleth, he, waw, zayin, complementary, and anticomplementary conjugates are given in C. Kimberling, "Collineations, Conjugacies, and Cubics," Forum Geometricorum 2 (2002) 21-32.
		isogonal conjugate Suppose P = p : q : r is a point not on a sideline of ABC.
		The vertices are the isogonal conjugates of the the vertices of the line-polar triangle of the points P
	faculty.evansville.edu /ck6/encyclopedia/glossary.html (4432 words)

	PlanetMath: isogonal conjugate
		Consider now their isogonals conjugates (reflections on the inner angle bisectors).
		An example of isogonal conjugate pair is found by looking at the centroid of the triangle and the Lemoine point.
		This is version 4 of isogonal conjugate, born on 2002-09-02, modified 2003-08-19.
	planetmath.org /encyclopedia/IsogonalConjugate.html (202 words)

	Q002
		the trilinear polar of P* and the orthotransversal of P* are parallel together with the circumcircle (P* is the isogonal conjugate of P).
		the isogonal conjugate of oc(P) lies on the line KP or, equivalently, oc(P) lies on the circumconic through G and P*.
		G, the O-isoconjugate of P, the isotomic conjugate of the isogonal conjugate of the barycentric square P^2 of P are collinear.
	perso.orange.fr /bernard.gibert/curves/q002.html (719 words)

	Springer Online Reference Works
		A circle is said to be isogonal with respect to two other circles if it makes the same angle with these two, [a1].
		The Gergonne point is the isotomic conjugate of the Nagel point.
		isogonal conjugation and isotomic conjugation, are better regarded as involutions of the projective plane
	eom.springer.de /I/i130080.htm (207 words)

	MathLinks Math Forums :: View topic - Isogonal conjugation with respect to a triangle
		Note that the isogonal conjugate of the point P with respect to triangle ABC is not defined if P is one of the three vertices of triangle ABC.
		If the isogonal conjugate of a point P with respect to triangle ABC is called Q, then the isogonal conjugate of this point Q with respect to triangle ABC is the point P again.
		The orthocenter H and the circumcenter O of triangle ABC are isogonal conjugate points with respect to triangle ABC.
	www.mathlinks.ro /Forum/viewtopic.php?t=18472 (2336 words)

	PlanetMath: fundamental theorem on isogonal lines
		This theorem is a direct consequence of Ceva's theorem (trigonometric version).
		"fundamental theorem on isogonal lines" is owned by drini.
		This is version 1 of fundamental theorem on isogonal lines, born on 2002-09-02.
	planetmath.org /encyclopedia/FundamentalTheoremOnIsogonalLines.html (83 words)

	K003
		It is the only isogonal pK60+ and the three asymptotes concur at G. These asymptotes meet the cubic at three finite points which lie on the line homothetic of the Brocard line OK under h(G,2/3).
		Locus of point P such that the antipedal triangle of P and the circumcevian triangle of its isogonal conjugate are perspective (together with the line at infinity and the circumcircle).
		Locus of point P such that the pedal triangles of P and its isogonal conjugate P* are orthologic (together with the line at infinity and the circumcircle).
	perso.orange.fr /bernard.gibert/Exemples/k003.html (1473 words)

	triangle centres
		the cyclocevian conjugate of Fi is the third meet of the cubic with LFi.
		the X69-Ceva conjugate of P is the third meet with PH (H the secondary pivot).
		It follows that the tangential of L is its cyclocevian conjugate X1032.
	www.maths.gla.ac.uk /~wws/cabripages/triangle/ninepoint/lucas.htm (727 words)

	Darij Grinberg
		First, the notion of isogonal lines or isogonals and the notion of isogonal conjugates with respect to a triangle are introduced.
		Proofs are given for the existence of the isogonal conjugate, for the fact that the isogonal conjugate of the circumcircle is the line at infinity, for the pedal circle theorem and for the relationship between the isogonal conjugate and reflections in sidelines, as well as for other properties.
		We continue with identifying the isogonal conjugates of the incenter, the excenters, the orthocenter, the circumcenter and the centroid of the triangle.
	de.geocities.com /darij_grinberg (3901 words)

	MathLinks Math Forums :: View topic - A property of isogonal conjugates in a triangle [SOLVED]
		Let P be a point in the plane of a triangle ABC, and P* its isogonal conjugate with respect to triangle ABC.
		Same way the reflection of P wrt BC is the isogonal conjugate of A wrt BP*C and your theorem.
		Similarly, if P* is the isogonal conjugate of the point P with respect to the triangle ABC, and A'B'C' is the precevian triangle of the point P with respect to the triangle ABC, then the line AB' is the harmonic conjugate* of the line AP* with respect to the angle CAB.
	www.mathlinks.ro /Forum/viewtopic.php?t=18423 (2079 words)

	Multidimensional Glossary
		As a rule, the facets of a conjugate polytope are conjugates of the facets of the first polytope, and the vertex figures of a conjugate polytope are conjugates of the vertex figures of the first polytope, whenever such conjugacies are apparent.
		Just as the two different isogonal complete pentagons may be distinguished by their adjunct rules, so may two different small and great complex icosidodecahedra be distinguished by which of the complete pentagons their vertex figures are.
		It corresponds to a uniform isochoric polychoron in the rox regiment whose cells are 120 identical small rhombidodekahedra (in a small rhombidodekahedron, the faces are the 30 squares of a rhombicosidodecahedron together with twelve decagons; its vertex figure is the butterfly quadrilateral).
	members.aol.com /Polycell/glossary.html (16070 words)

	Art of Problem Solving Forum (Site not responding. Last check: 2007-10-15)
		Let P and Q be two mutually isogonal conjugate points with respect to a triangle ABC, and let E be an arbitrary point on the line CQ.
		Then, of course, P is again the isogonal conjugate of Q with respect to triangle ABC; hence, the relation of the points P and Q is mutual, and we can say the points P and Q are mutually isogonal conjugate points.
		Since the points P and P' are isogonal conjugates with respect to triangle ABC, the lines AP and AP' are symmetric with respect to the angle bisector of the angle CAB.
	www.artofproblemsolving.com /Forum/viewtopic.php?t=6082 (2221 words)

	ENCYCLOPEDIA OF TRIANGLE CENTERS
		X(9) is the internal center of similitude of the Bevan circle and Spieker circle; the external center is X(1706).
		X(10) is the internal center of similitude of the Apollonius and nine-points circles.
		X(43) is the external center of similitude of the Bevan circle and Apollonius circle; the internal center is X(1695).
	faculty.evansville.edu /ck6/encyclopedia/ETC.html (10344 words)

	isogons
		We often say that one is the isogonal reflection of the other, but it should be clear that if L2 is the isogonic reflection of L1, then L1 is the isogonic reflection for L2. Two points on these rays, such as X and Y, are called
		One famous pair of isogonal conjugates is the orthocenter (intersection of the altitudes) and the circumcenter (center of the circle which circumscribes a triangle).
		What is perhaps more amazing is that a similar situation exists for any two isogonal conjugates X and Y. The circumcircle of the pedal triangle of X is also the circumcircle of the pedal triangle of Y, and the center of the circle is the midpoint of the segment XY.
	www.pballew.net /isogon.html (1694 words)

	Simson Line From Isogonal Perspective
		The applet below illustrates a property of isogonal conjugates that generalizes the Simson line.
		The six feet of the perpendiculars to the sides of triangle ABC dropped from a point P and its isogonal conjugate Q are concyclic, i.e.
		The pedal circles of a point and its isogonal conjugate coincide.
	www.cut-the-knot.org /Curriculum/Geometry/IsoSimpson.shtml (425 words)

	Small Vocabulary
		Take a point P and the isogonal lines of the cevians through P. These isogonals intersect at a point Q. The correspondence P->Q is the isogonal transformation (it is an involution).
		The isogonal transformation is a special kind of quadratic transformation.
		to a triangle ABC: take the isogonal Q of P and the six projections of P and Q on the sides of ABC: they lie on a circle: the pedal circle of P w.r.
	www.euclidraw.com /Eng_fls/EUC_htmls/Vocabulary.htm (4232 words)

	4-D Snubs
		Above are sections of sadi - the only uniform convex 4-D snub, rasdi - its spiky conjugate, and rappisdi - the head of a 3
		Also shown are two views of the vertex figure of the snub polychoron "rapsady" - the only known uniform
		There are many isogonal snubs, but there are two many conditions necessary to make them uniform, trying to make one of the
	hometown.aol.com /hedrondude/snubs.html (331 words)

	Read This: Essays on Numbers and Figures
		At their best, the essays pose a question or state an interesting result, and then proceed to explore in an intuitive way some of the consequences leading from the initial topic.
		The first essay, Conjugate Numbers, begins by noticing that 2 plus the square root of 3, raised to increasing powers, gets closer and closer to an integer value.
		The essay on isogonal conjugate points covers seven theorems in the space of five pages.
	www.maa.org /reviews/prasolov.html (1180 words)

	Springer Online Reference Works
		The theory of curves in the conformal plane is constructed from the theory of sequences of circles: each curve is regarded as an envelope, namely the sequence with invariant
		is constant, then the curve turns out to be the isogonal trajectory of a pencil of circles, that is, a loxodrome.
		, defining the pencil of the circles, conjugate to the pencil
	eom.springer.de /C/c024740.htm (550 words)

	nrich.maths.org::Mathematics Enrichment::NRICH (Site not responding. Last check: 2007-10-15)
		This is because they are closely related points - the technical term is that they are "isogonal conjugates" of each other.
		David, I'm not familiar with the term "isogonal conjugates".
		The point P' is then said to be the isogonal conjugate of the original point P. (Every point in the plane of a triangle has an isogonal conjugate unless it lies on the circumcircle, in which case the reflected cevians will be parallel and the point P' recedes to infinity).
	www.nrich.maths.org /askedNRICH/edited/3552.html (1453 words)

	Isogonal Conjugates
		The Isogonal Conjugates of a point, P, is a point P' at the intersection of the cevians you get by reflecting each cevian about the bisector of the angle of the original triangle.
		The orthocenter (the common intersection of the three altitudes of a triangle, or their extensions, which must meet in a point) and the circumcenter (the center of the circumscribing circle) are isogonal conjugates of one another.
		That is, if you have a set of cevians concurring in a point P, the new set of cevians you get by reflecting each cevian about the bisector of the angle of the original triangle will also concur at a new point P'.
	mcraefamily.com /mathhelp/GeometryTriangleIsogonalConjugates.htm (434 words)

	[No title]
		The point X(55) is the internal center of similtude of the circumcircle and the incircle of triangle ABC, also known as the isogonal conjugate of the Gergonne point.
		The isogonal conjugate of the Bevan-Schröder point Sb is the point X(1320); this point lies on the line NF, where N is the Nagel point and F the Feuerbach point of triangle ABC.
		the isogonal conjugate of the Mitten point) is poristically fixed, the orthocenter X(65) of the Gergonne triangle XYZ is poristically fixed, and the centroid X(354) of the Gergonne triangle XYZ is poristically fixed.
	de.geocities.com /darij_grinberg/Schroeder/Schroeder.html (6625 words)

	Short-Latency Disparity Vergence in Humans -- Busettini et al. 85 (3): 1129 -- Journal of Neurophysiology
		Compared with the isogonal components, the orthogonal components were small and slow to develop, so that directional errors
		2 was fitted to the mean isogonal responses obtained with horizontal and vertical disparity steps using patterns with dots of various sizes
		Regarding the best-fit isogonal parameters, the only one that consistently showed a systematic dependence on dot size was
	jn.physiology.org /cgi/content/full/85/3/1129 (9726 words)

	Isogonal_Conjugation (Site not responding. Last check: 2007-10-15)
		The resulting lines EA, EB, EC concur at a point E, called the [Isogonal-Conjugate] of D. The map f: D --- > E is the [isogonal conjugation], defined by the triangle t.
		The most well known pair of conjugate points are the pairs of (circumcenter O, orthocenter H), (centroid, symmedian) of the triangle.
		-The six projections of D and its isogonal E on the sides lie on a circle of radius r (the circumcircle of the triangle with vertices I, J, H, the projections of D on the sides of t).
	www.math.uoc.gr /~pamfilos/eGallery/problems/Isogonal_Conjugation.html (240 words)

	Uniform Polychora (Site not responding. Last check: 2007-10-15)
		In 2002 I found Iquipadah and it's conjugate Gaquipadah, and later realized that all non-prime dimensions have iquipadah like figures.
		It wasn't until a year later, when I was looking at sidpith, that I noticed that there was a uniform compound of two sodips (square-octagon duoprisms) inside and it was blendable with sidpith and lead to a new uniform polychoron, of course it also had a conjugate in the gittith regiment.
		The idcossids are based off of the 10-padohi compound, were the verfs are facetings of a pentagonal antipodium duo-combo.
	members.cox.net /hedrondude/polychora.shtml (3919 words)

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