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Topic: Nine point circle


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 Nine-point circle - Wikipedia, the free encyclopedia
The radius of any nine-point circle is half the length of the radius of the circumcircle of the corresponding triangle.
In geometry, the nine-point circle is a circle that can be constructed for any given triangle.
He discovered the six point circle, recognizing the significance of points the midpoints of the three sides of the triangle and the feet of the altitudes of that triangle.
en.wikipedia.org /wiki/Nine_point_circle   (603 words)

  
 PlanetMath: nine-point circle
The nine point circle also known as the Euler's circle or the Feuerbach circle is the circle that passes through the feet of perpendiculars from the vertices
Property 4 : The center of the nine-point circle is the midpoint of the line segment joining the orthocenter and the circumcenter, and hence lies on the Euler line.
This is version 3 of nine-point circle, born on 2002-12-02, modified 2003-01-15.
planetmath.org /encyclopedia/EulerCircle.html   (178 words)

  
 Nine Point Circle
The nine point circle is the circumcirlce of the orthic triangle H
of the segments connecting the orthocenter with vertices, lie on a circle, known as the 9 point circle.
Since the triples come in pairs that comprise six concyclic points, each pair of the triples define the same circle, which exactly means that all three triples - 9 points in all - all lie on the same circle.
www.cut-the-knot.org /Curriculum/Geometry/SixPointCircle.shtml   (477 words)

  
 Nine point circle: Facts and details from Encyclopedia Topic
The nine point circle is tangent externally to the three excircle[For more facts and a topic of this subject, click this link]s and tangent internally to the incircle[Follow this hyperlink for a summary of this subject] of the triangle, EHandler: no quick summary.
Terquem was the first to use the name nine point circle (as he was the first to associate nine special points with the circle).
Olry terquem (1782 - 1862) was a french mathematician, best known for his work in geometry, where he proved the feuerbachs theorem about the nine...
www.absoluteastronomy.com /encyclopedia/n/ni/nine_point_circle.htm   (966 words)

  
 Euler's line - Wikipedia, the free encyclopedia
The center of the nine-point circle lies midway between the orthocenter and the circumcenter, and the distance from the centroid to the circumcenter is half that from the centroid to the orthocenter.
In geometry, Euler's line (red line in the image), named after Leonhard Euler, 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.
Leonhard Euler showed that in any triangle, those four points are collinear.
en.wikipedia.org /wiki/Euler's_line   (119 words)

  
 The History of the Nine-Point Circle
The nine-point circle, also known as Euler’s circle, is the circle that passes through the feet of the perpendiculars dropped from the vertices of any triangle on the sides opposite them.
tangent to the three sides of the triangle; it is internally tangent to the inscribed circle and externally tangent to each of the circles which touch the sides of the triangle externally.”  The point where the incircle and the nine-point circle touch is now called the Feuerbach point (Feuerbach, 1).
The earliest date for the discovery of the nine-point circle was in 1804 by Bevan whose theorem appears in Mathematical Repository Vol.
www.sienahts.edu /~km105427/circle.htm   (572 words)

  
 Nine Point Circle Investigation
Both the orthocenter and the circumcenter lie on Euler's Line and the center of the nine point circle is related to these two points by dilation.
On your sketch, construct a segment from the orthocenter to the nine point center and measure it's length.
The points where the altitudes intersect the sides of the triangle are called the feet of the altitudes.
www.geom.uiuc.edu /~demo5337/Group2/nineptcircle.html   (890 words)

  
 New Page 3
He also proved that the nine point circle touches the inscribed circle as well as the three excircles, which are the circle that are tangent to the sides of the triangle.
This property of the nine point circle was published in his paper in 1822.
Not to say that Feuerbach was the first and last to do work on the nine point circle, there were also others who shared his interests: Brianchon and Poncelet also published a proof of this theorem a year before Feuerbach did.
www.math.ubc.ca /~cass/courses/m309-01a/shah/history.htm   (234 words)

  
 klein view page
the circumcentre, the nine-point centre, and the Fermat points.
the same proof shows that there is a circle through the first two points, the symmedian point and the centre
The Parry circle is remarkable in that it occurs in the families of the
www.maths.gla.ac.uk /~wws/cabripages/misc/misc0.html   (872 words)

  
 The Nine-point Circle
Feuerbach's Theorem, published in 1822, states that the incircle and the nine-point circle are tangent at a point typically called the Feuerbach point.
In addition, the nine-point circle is tangent to the three excircles.
Its center is the midpoint of the segment joining the orthocenter and the circumcenter of the triangle, and consequently lies on the Euler line.
www.math.sunysb.edu /~scott/mat360.spr04/cindy/ninepoint.html   (187 words)

  
 NINE-POINT CENTER
As you see in the sketch, a circle passes through all nine of the points D,E,F,G,H,I,J,K,L. It is the nine-point circle of triangle ABC, and its center, N, is the nine-point center.
faculty.evansville.edu /ck6/tcenters/class/npcenter.html   (84 words)

  
 nine-point-circle
Given any (Euclidean) triangle the circle which passes through the three points which are the feet of the perpendiculars to the sides of the triangle also passes through the midpoints of the sides of the triangles, as well as the points which bisect the segments joining the three vertices to the orthocenter of the triangle!
This circle is generally known as the 9-point circle.
The centroid of a triangle is the point where the three medians of the triangle meet.
www.york.cuny.edu /~malk/mycourses/math244/nine-point-circle.html   (291 words)

  
 Homework Problems 15
Thus, angles FMD and A are supplementary, implying that M is on the circle determined by A, F and D. Show that, on the Euler line, the centroid and the orthocenter divide internally and externally in the same ratio the segment whose endpoints are the circumcenter and the center of the nine-points circle.
Since NL is a radius of the nine-point circle, and OK is a radius of the circumcircle, their ratio is 1:2 (see the statement of exercise 8).
Join K to H (the orthocenter) and let L be the point where this line segment meets the nine-point circle.
www-math.cudenver.edu /~wcherowi/courses/m3210/hghw15.old   (739 words)

  
 AoPS Math Forum :: View topic - radius of apollonius circle
The center N of the nine-point circle is the circumcenter of the triangle A'B'C'.
The nine-point circle of triangle ABC is the circumcircle of triangle A'B'C', where A', B', C' are the midpoints of the sides BC, CA, AB.
On the other hand, the nine-point circle, which touches the three excircles externally, is mapped by this inversion to a circle which touches the three excircles internally (i.
www.artofproblemsolving.com /topic-5958.html   (1161 words)

  
 Nine Point Circle
The midpoints of the three sides of the triangle ABC as illustrated below with points M, N, and L. To construct the Nine-Point Circle a triangle is formed from the three midpoints and the intersection of the perpendicular bisectors forms the center of the Nine-Point Circle.
The Nine Point Circle illustrated above is constructed from and triangle and three different sets of points relating to the triangle.
The Nine-Point Circle Theorem-The radius of the Nine-Point Circle is one-half the circumradius of triangle ABC.
web.pdx.edu /~dgreene/9pointcircle.htm   (289 words)

  
 spheres.html
Since the circles have three points in common, they're equal and they're the nine point circle of the triangle.
It's remarkable that nine particular triangle points lie on one and the same circle because we know that three noncollinear points determine a circle and if there are four or more points given then in general they don't lie on the same circle.
The nine point circle is a circle passing through the points K, L, M, P, Q, R, X, Y and Z.
republika.pl /mtkacz/spheres.html   (1928 words)

  
 Math Forum - Ask Dr. Math
The center of the nine point circle, also called the nine point center, is a very well-known triangle center.
You can read about the Euler line in our archives: Euler Line http://mathforum.org/dr.math/problems/christen6.8.98.html A very nice result on the nine point circle is proven by Feuerbach in 1822: it is tangent to the incircle and the excircles of a triangle.
Date: 02/28/99 at 06:18:05 From: Doctor Floor Subject: Re: Nine-point Circle This is a very famous triangle problem that was presented by the Swiss mathematician Euler in 1765.
mathforum.org /library/drmath/view/55091.html   (431 words)

  
 The nine point circle
Then the nine points A', B', C', a, b, c, D, E and F all lie on a circle.
The circle drawn is the circle with center the midpoint of A'a and through A'.
ABC, let A', B' and C' be the midpoints of the opposite sides; let D, E and F be the feet of the altitudes; let H be the orthocenter, and let a, b and c be the midpoints of AH, BH and CH respectively.
www.nevada.edu /~baragar/geom/Ninept.htm   (145 words)

  
 The nine-point circle
Since the three points K, L and M are diametrically opposite to A', B' and C' the two triangles are symmetric with respect to the center of the nine-point circle N.
The feet of the three altitudes (D, E and F) of any triangle, the midpoints of the three sides (A', B' and C'), and the midpoints of the segments from the three vertices to the orthocenter (K, L and M), all lie on the same circle, of radius
It is the midpoint of the segment connecting the orthocenter and circumcenter.
www.math.uci.edu /~mathcirc/math194/lectures/advanced3/node1.html   (240 words)

  
 Cantor's Theorem
The point of concurrence is known as Cantor's point of the polygon.
In complex variables (or affine geometry, where we can add points) the proof is truly simple.
The lines in the theorem are known as Cantor's lines and their common point is sometimes referred to as the Cantor's point.)
www.cut-the-knot.org /Curriculum/Geometry/CantorPoint.shtml   (268 words)

  
 TETRAHEDRON - LoveToKnow Article on TETRAHEDRON
The " twelve-point sphere," discovered by P. Prouhet (1817-1867) in 1863, is somewhat analogous to the nine-point circle of a triangle.
This theorem has been generalized for any tetrahedron; a sphere can be drawn through the four feet of the perpendiculars, and consequently through the mid-points of the lines from the vertices to the centre of the hyperboloid having these perpendiculars as generators, and through the orthogonal projections of these points on the opposite faces.
If (, v, w, t) be the co-ordinates of any point, then the relation ++'+/= R, where R is a constant, invariably holds.
www.1911ency.org /T/TE/TETRAHEDRON.htm   (318 words)

  
 More Unusual Properties of Acute Triangles
The tangents to the Nine-Point Circle at the midpoints L, M, and N of the sides of the triangle form a triangle, RST, that is similar to the orthic triangle (the triangle DEF).
The Nine-Point Circle of a triangle "touches" the incircle and the three excircles.
MacKay, J. History of the Nine Point Circle.
www.world-destiny.org /or/moretri.htm   (160 words)

  
 THE LIVING LIBRARY
Any sequence of numbers can be reduced with digit sums and shown on the nine point circle: the sequence of square numbers, cube numbers, triangular numbers, the Fibonacci sequence and so on all have interesting patterns as do the digits in recurring decimal expansions of fractions.
Thus, though it is a very simple device, the nine point circle gives us new ways of looking at numbers and their relationships and meaning.
If you are interested in the philosophical aspect of numbers the nine numbers can represent the steps in the creative process (on an individual or cosmic level) where one represents the creator and nine the full creation, with all the stages of the creative process in-between.
www.sacredscience.com /LivingLibrary/messages/293/425.html?1023562882   (2254 words)

  
 writeup4
Can a circle be constructed using any arbitrary triangle containing the midpoints of each side of the triangle, the points where the altitudes meet the sides of the triangle, and the midpoints of the segments on the altitudes between the vertex and the orthocenter?
Construct a circle using O as its center and OM as the radius.
Label the point where they intersect as H. Label the points where they intersect the sides of the triangle as R, S, and T for sides AB, BC, and AC respectively.
www.auburn.edu /~blackjw/writeup4.html   (252 words)

  
 Nine Point Circle
Amazingly, these nine points all lie on the same circle.
As you will see, the other six points you didn't select are also on this circle.
Pick any three of these points and draw a circle through them.
www.math.psu.edu /dlittle/java/Geometry/Euclidean/ninepointcircle.html   (76 words)

  
 Lesson on Nine Point Circle and Euler Line
Students know what happens to the nine points of the circle when a triangle is isosceles.
Students know how the orthocenter, centroid, circumcenter, and center of nine-point circle are position with each other the nine point circle, and the Euler line.
Geometer's Sketchpad is an excellent software for this lesson because it enables students to examine many cases of the nine-point circle and Euler line, thus increasing students' ability to formulate and explore conjectures on them.
www.bsu.edu /web/mdlade/Finallesson   (489 words)

  
 The Nine Point Circle
The Nine-Point circle for any triangle passes through the three mid-points of the sides, the three feet of the altitudes, and the three mid-points of the segments from the respective vertices to orthocenter.
The final three points of the nine point circle lie on the midpoints of the segments NA, NB, and NC.
In this special case, three of the nine points in the nine point circle appear to dissappear.
web.pdx.edu /~blaesing/assignment2.html   (749 words)

  
 The Geometer's Sketchpad® - : Nine Point Circle
The Nine Point Circle, and sketches like it, will be vastly more useful in DR2, which will support labels for objects.
In any triangle, one can locate a circle that passes through the midpoints of that triangle's edges, the midpoints of its perpendicular segments, and the feet of a given triangle's perpendiculars.
In this sketch, drag the vertices of the yellow triangle to alter the triangle.
www.ipst.ac.th /it/javasketcdpad/jsp_demo_9pts.htm   (158 words)

  
 Incidence in Feuerbach's Theorem: What Is It About?
The point of tangency of the incircle and the 9-point circle is known as the Feuerbach point N. Let the base triangle be ABC, and denote the points of tangency in question F
According to Feuerbach's theorem, the 9-point circle of a triangle touches its incircle and the three excircles.
The point is X(12) in Kimberling's Encyclopedia of Triangle Centers, a harmonic conjugate of X(11), the Feuerbach Point F
www.cut-the-knot.org /Curriculum/Geometry/FeuerbachIncidence.shtml   (173 words)

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