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Topic: Orthocenter

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	Orthocenter
		The orthocenter of a triangle is the point where the three altitudes meet. This point may be inside, outside, or on the triangle. Here are some properties of the orthocenter that I find to be interesting.
		It is well known that the orthocenter is an isogonal conjugate of the circumcenter. This heavy language means that if the altitudes are reflected in the angle bisectors from the same vertex, the three new lines intersect at the circumcenter of the triangle.
		In each case the conics have their foci at the orthocenter and the circumcenter, and the conic center is at the center of the nine point circle.
	www.pballew.net /orthocen.html (1819 words)

	PlanetMath: orthocenter
		The orthocenter of a triangle is the point of intersection of its three heights.
		Orthocenter is one of the most important triangle centers and it is very related with the orthic triangle (formed by the three height's foots).
		This is version 3 of orthocenter, born on 2001-10-31, modified 2002-03-09.
	www.planetmath.org /encyclopedia/Orthocenter.html (122 words)

	Triangle - Wikipedia, the free encyclopedia
		The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter.
		Euler's line is a straight line through the centroid (yellow), orthocenter (blue), circumcenter (green) and center of the nine point circle (red).
		The centroid (yellow), orthocenter (blue), circumcenter (green) and center of the nine point circle (red point) all lie on a single line, known as Euler's line (red line).
	www.wikipedia.org /wiki/Triangle (2072 words)

	Travels with an Orthocenter (Site not responding. Last check: 2007-11-04)
		The red curve is the locus of the orthocenter of a triangle with base AB between - 5 and 5 on the x-axis and vertex C moving along the cosine curve.
		The orthocenter approaches the origin, where it is undefined, as the third vertex of the triangle approaches an infinite length from the base.
		For example, we began with the locus of the orthocenter as the third vertex is moved along a line parallel to the base.
	jwilson.coe.uga.edu /Texts.Folder/ortho/Orthotravels.html (718 words)

	Altitude (triangle) Summary
		The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter.
		The isogonal conjugate of the orthocenter is the circumcenter.
		It is the pedal triangle of the orthocenter of the original triangle.
	www.bookrags.com /Altitude_(triangle) (622 words)

	All about altitudes
		BCH, while B and C are the orthocenters of triangles ACH and ABH, respectively.
		H is an angle bisector in the orthic triangle.
		The argument that shows that three points - the circumcenter O, the centroid M, and the orthocenter H - lie on the same line is reversible.
	www.cut-the-knot.org /triangle/altitudes.shtml (1378 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-04)
		Because the incenter is the point at which the bisectors of the three angles of the triangle meet, the incenter is necessarily inside the triangle.
		orthocenter the first component is from Greek orthos "straight, upright," hence "perpendicular, from the Indo-European root wrodh- "to grow straight, upright"; the second component is center.
		With regard to a triangle, the place where the three altitudes (which are perpendicular to the sides) meet is called the orthocenter...
	www.mathforum.com /library/drmath/view/55422.html (406 words)

	Encyclopedia: Triangle (Site not responding. Last check: 2007-11-04)
		In geometry, Thales theorem (named after Thales of Miletus) states that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle.
		A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas.
		In geometry, Eulers line (red line in the image) is the line passing through the orthocenter (blue), the circumcenter (green), the centroid (yellow), and the center of the nine point circle (red point) of any triangle.
	www.nationmaster.com /encyclopedia/triangle (4215 words)

	Sketchpad Investigations of the Orthocenter (Site not responding. Last check: 2007-11-04)
		orthocenter - the intersection of the three altitudes of a triangle.
		Drag the triangle vertices, and observe the position of the orthocenter.
		This investigation was first prepared as a presentation for a Northwest Mathematics Interactions workshop, in the fall of 2000.
	www.whistleralley.com /orthocenter/orthocenter.htm (813 words)

	Altitude (triangle) (Site not responding. Last check: 2007-11-04)
		The orthocenter liesinside the triangle (and consequently the feet of the altitudes all fall on the triangle) if and only if the triangle is notobtuse (i.e.
		The orthocenter, along with the centroid, circumcenter and center of the nine pointcircle all lie on a single line, known as Euler's line.
		The center ofthe nine point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroidand the circumcenter is half that between the centroid and the orthocenter.
	www.therfcc.org /altitude-triangle--137136.html (304 words)

	M3210 Sample Exam II (Site not responding. Last check: 2007-11-04)
		the three midpoints of the sides of the triangle, the three feet of the altitudes of the triangle and the three midpoints of the segments drawn from the vertices to the orthocenter of the triangle.
		Prove that the internal bisectors of two angles of a triangle and the external bisector of the third angle intersect the opposite sides of the triangle in three collinear points.
		The orthocenter is the point where the three altitudes of the triangle meet.
	www-math.cudenver.edu /~wcherowi/courses/m3210/hgex2sam.html (623 words)

	Assignment2
		The orthocenter (H) of a triangle is the intersection of the three lines containing the altitudes.
		The figure below is a construction using Geometer's Sketchpad of a triangle with the three mid-points of the sides, the altitudes, the feet of the altitudes, the orthocenter H, and the three mid-points of the segments from the respective vertices to the orthocenter.
		Notice that, with a right triangle, the orthocenter, two of the feet of the altitudes, and the midpoint between the orthocenter and the vertex of the right angle are concurrent with the vertex of the right angle.
	web.pdx.edu /~pconnor/Assignment2.html (780 words)

	Nine point circle (Site not responding. Last check: 2007-11-04)
		It is named so because it passes through nine significant points, with six of them lying onthe triangle itself: the midpoints of the three sides, the feet of the altitudes, and the midpoints of the portion of altitude between the vertices and the orthocenter.
		Soonafter Feuerbach, mathematician Olry Terquem also proved what Feuerbach didand added the three points that are the midpoints of the altitude between the vertices and the orthocenter.
		The center of the nine point circle (the nine point center) lies on the triangle's Euler line, at the midpoint between the triangle's orthocenter andcircumcenter.
	www.therfcc.org /nine-point-circle-70716.html (425 words)

	Altitude Manipulator (Site not responding. Last check: 2007-11-04)
		Note: there are many right-triangles with the orthocenter coinciding with a base endpoint, even when the base is of a fixed length, as in this figure.
		There is no triangle solution when the orthocenter is on the line containing the base.
		The triangle is degenerate when the orthocenter is on a perpendicular to the base, through a base endpoint.
	www.math.clemson.edu /~rsimms/triangle/altitude_manipulator.html (147 words)

	The Euler Line
		Centroid is always located between the circumcenter and the orthocenter twice as close to the former as to the latter.
		ABC and the midpoints of segments that join the orthocenter with the vertices of the triangle.
		Therefore, the quadrilateral formed by the orthocenters of the four triangles is the reflection of x
	www.cut-the-knot.com /triangle/EulerLine.shtml (562 words)

	Orthocenter of a triangle - Math Open Reference
		The orthocenter of a triangle is the point where its altitudes intersect.
		The orthocenter is not always inside the triangle.
		It is possible to construct the orthocenter of a triangle using a compass and straightedge.
	www.mathopenref.com /triangleorthocenter.html (197 words)

	Problem 1
		Inscribe a triangle ABC in circle l such that its orthocenter is point H, and side BC is congruent to segment a, where a is given.
		Being that AH=2OD holds on the basis of the construction, using the Lemma, it follows that the point A lies on the circle l and H is the orthocenter.
		It follows that the triangle ABC with the orthocenter H is inscribed in circle l.
	www.matf.bg.ac.yu /~daad/work/usegclc/example1.htm (673 words)

	The Complete Quadrilateral
		Next we consider the four triangles formed by the four lines (omitting one of them at a time.) The orthocenters of the triangles are collinear and the line (Ortholine in the applet) is perpendicular to the line (Midline in the applet) of the three mid-diagonals.
		Also, the ortholine serves as the common radical axis of the three circles constructed on the diagonals as diameters, such that whenever the circles intersect, all three of them intersect in two points on the ortholine.
		For the centroid, the circumcenter and the orthocenter it is quite clear that if the lines intersect on BD for one position of C, then the same is true for all other positions as well.
	www.maa.org /editorial/knot/CompleteQuadrilateral.html (1548 words)

	Sketchpad Investigations of the Orthocenter
		orthocenter - the intersection of the three altitudes of a triangle.
		Drag the triangle vertices, and observe the position of the orthocenter.
		This investigation was first prepared as a presentation for a Northwest Mathematics Interactions workshop, in the fall of 2000.
	whistleralley.com /orthocenter/orthocenter.htm (813 words)

	Altitude (triangle) - Wikipedia, the free encyclopedia
		The isogonal conjugate of the orthocenter is the circumcenter.
		Four points in the plane such that one of them is the orthocenter of the triangle formed by the other three are called an orthocentric system or orthocentric quadrangle.
		Triangle centers by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
	www.wikipedia.org /wiki/Altitude+(triangle) (371 words)

	Altitude of a Triangle, Orthocenter
		Formally, the shortest line segment between a vertex of a triangle and the (possibly extended) opposite side.
		Note: Every triangle have 3 altitudes which are concurrent, intersecting at one point - orthocenter.
		For an obtuse triangle (having one angle exceeding 90°), the orthocenter lies outside the triangle, because the lines of the altitudes intersect in one point.
	www.math10.com /en/geometry/altitude.html (102 words)

	assignment 2 webpage
		The nine points on the circle are found using midpoints, altutiudes and the orthocenter.
		Finally the last three points are the midpoints of the segments from the orthocenter to each vertex.
		The perpendicular bisectors intersect at the circumcenter labeled point, O. The circumcenter is also on the same line as the orthocenter, centroid, and the center of the circle.
	web.pdx.edu /~kelseyk/assignment2.html (569 words)

	Homework Problems 13 (Site not responding. Last check: 2007-11-04)
		Use the notation of Figure 4.3 to name all of the triangles whose vertices are three of the given points and whose orthocenter is the fourth distinct point.
		They meet at the right angle, so it must be the orthocenter.
		The altitudes drawn to the sides of the obtuse angle always lie outside the triangle.
	www-math.cudenver.edu /~wcherowi/courses/m3210/hghw13.html (436 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-04)
		Date: 01/06/97 at 01:51:53 From: Doctor Pete Subject: Re: Euler line The Euler line of a triangle is the line which passes through the orthocenter, circumcenter, and centroid of the triangle.
		The orthocenter is the intersection of the triangle's altitudes.
		The circumcenter is the center of the circumscribed circle (the intersection of the perpendicular bisectors of the three sides).
	www.mathforum.org /dr.math/problems/beck1.5.97.html (224 words)

	The point of intersection of the altitudes of a triangle is called ... - BlurtIt
		The three altitudes intersect in a single point, called the orthocenter of the triangle.
		The orthocenter lies inside the triangle if and only if the triangle is acute.
		The three vertices together with the orthocenter are said to form an orthocentric system.
	www.blurtit.com /q384327.html (180 words)

	Activity 9
		Orthocenter An altitude of a triangle is a segment that is drawn so that it passes through a vertex and is perpendicular to the opposite side.
		The intersection of the three altitudes of a triangle is called the orthocenter.
		To draw an altitude that lies outside the triangle, it is necessary to extend two of the sides of the triangle.
	homepage.mac.com /efithian/Geometry/Activity-09.html (762 words)

	SparkNotes: Geometric Theorems: Theorems for Segments within Triangles
		The lines containing the altitudes of a triangle meet at one point called the orthocenter of the triangle.
		Because the orthocenter lies on the lines containing all three altitudes of a triangle, the segments joining the orthocenter to each side are perpendicular to the side.
		This means that the orthocenter isn't necessarily in the interior of the triangle.
	www.sparknotes.com /math/geometry2/theorems/section2.rhtml (784 words)

	PlanetMath: Euler line proof
		Repeating the same argument for the other medians prove that
		lies on the three heights and therefore it must be the orthocenter.
		Cross-references: ratio, lies on, argument, height, triangles, midpoint, median, orthocenter, centroid
	www.planetmath.org /encyclopedia/EulerLineProof.html (70 words)

	The Euler Line of a Triangle
		For an acute triangle, the orthocenter lies inside the triangle; for an obtuse triangle, it lies outside the triangle; and for a right triangle, it coincides with the vertex at the right angle.
		Since the altitudes of the original triangle meet at the orthocenter H of the original triangle, the altitudes of the medial triangle will meet at its orthocenter H' which you can see in the figure is labelled O.
		This orthocenter O of the medial triangle is the circumcenter of the original triangle!
	aleph0.clarku.edu /~djoyce/java/Geometry/eulerline.html (929 words)

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