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# Topic: Symmedian point

###### In the News (Sat 17 Aug 19)

 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 Gergonne point of a triangle is the same as the symmedian point of the triangle's contact triangle. The symmedian point is the isogonal conjugate of the triangle's centroid. en.wikipedia.org /wiki/Symmedian   (214 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)

 AoPS Math Forum :: View topic - triangle and Lemoin point   (Site not responding. Last check: 2007-10-29) Symmedians connect the triangle vertices with the poles of the opposite sides with respect to the circumcircle. Hence, the symmedian point of triangle ABC is the point The point D is the point where the tangents to the circumcircle of triangle ABC at the points B and C meet. www.artofproblemsolving.com /Forum/post-187405.html   (682 words)

 Darij Grinberg Abstract: The Kosnita point of a triangle is the isogonal conjugate of the nine-point center. The Mitten point of a triangle, defined as the perspector of the medial and excentral triangles, is shown to be the radical center of a variable circle triad. 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. de.geocities.com /darij_grinberg   (3178 words)

 Bride's Chair The center of homothety is known as Lemoine's point or (in Germany) as Grebe's point. Point M therefore serves as the center of the square constructed on AB inwardly to Similar isosceles triangles on the sides of a given triangle ABC are the subject of Kiepert's theorem that asserts that the outer apexes of the Kiepert triangles form a triangle perspective to www.cut-the-knot.com /ctk/BridesChair.shtml   (992 words)

 isogons The Gergone point is the intersection of the segments from a vertex to the tangent point of the incircle with the opposite side. It is frequently called the Second Lemoine circle, and sometimes the Cosine circle because the length of the chords formed by two points of intersection nearest a vertex (EF for example) is proportional to the cosines of the angles (angle B for EF) at the adjacent vertices.. 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. www.pballew.net /isogon.html   (1694 words)

 Antiparallels   (Site not responding. Last check: 2007-10-29) Moving M to the intersection L of the symmedian with the tangent at A, we see that the other tangent, at B, passes also from L. In fact, points M, F the center of the circle {ABD} and L coincide in this case. 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)

 klein view page the circumcentre, the nine-point centre, and the Fermat points. There are many points associated with a triangle, such as the incentre and circumcentre. In the table, the letters e,f,b indicate that the points lie on the Euler line or on the Fermat or Brocard axis. www.maths.gla.ac.uk /~wws/cabripages/misc/misc0.html   (872 words)

 Mathh 114 Lab 1   (Site not responding. Last check: 2007-10-29) 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 when played with one ordering of the three points produces the positive Brocard circles and corresponding point, and which when played with the other ordering produces the negative Brocard circles and corresponding 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)

 Math Forum - Ask Dr. Math   (Site not responding. Last check: 2007-10-29) 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)

 Symmedian   (Site not responding. Last check: 2007-10-29) 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). K coincides with the Gergonne point of the tangential triangle A'B'C' of ABC. 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)

 [A] treatise of the analytical geometry of the point, line, circle and conic sections : containing an account of its ...   (Site not responding. Last check: 2007-10-29) Isogonal conjugates of inverse points with respect to circumcircle are twin points. Twin points are at the extremities of a diameter of circumequilateral hyperbola, Polar of Symmedian point is the same with respect to circumcircle and harmonic polygon, math-sahel.ujf-grenoble.fr /LiNuM/TM/Gallica/S099630.html   (2486 words)

 STEINER POINT   (Site not responding. Last check: 2007-10-29) The Steiner point of a triangle ABC is constructed as follows: First, let O be the circumcenter and K the symmedian point of ABC. The circle having segment OK as diameter is the Brocard circle. The line through O perpendicular to line BC passes through the Brocard circle in another point, a; similarly, obtain points b and c. faculty.evansville.edu /ck6/tcenters/class/steiner.html   (238 words)

 AoPS Math Forum :: View topic - Find the point!   (Site not responding. Last check: 2007-10-29) Hence, the point P is the centroid of triangle A'B'C' if and only if the point P is the centroid of triangle XYZ. Now, the triangle XYZ is the pedal triangle of the point P with respect to triangle ABC, and it is quite well-known that the point P is the centroid of triangle XYZ if and only if the point P is the symmedian point of triangle ABC (see, e. be a point in the interior of the triangle www.artofproblemsolving.com /Forum/post-138626.html   (766 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] www.onelook.com /?w=symmedian   (86 words)

 MathLinks Math Forum :: View topic - an easy problem!!!!   (Site not responding. Last check: 2007-10-29) intersect at the point with an equilateral pedal triangle, i.e., at the 1st isodynamic point An internal angle bisector cuts the opposite side internally in the ratio of the adjacent sides, an external angle bisector cuts the opposite side externally in this ratio, and a symmedian cuts the opposite side in the square of this ratio. Thus this point is either the radical center of the circles www.mathlinks.ro /Forum/post-200669.html   (654 words)

 circumcenter-2 Steiner inellipse, Steiner circumellipse, symmedian point, centroid, Tarry point, Steiner point Fermat points, Napoleon points, ninecenter, symmedian point, Kiepert hyperbola, Evans conic, autopolar triangle Nagel point, orthocenter, Mittenpunkt, De Longchamps point, point X(57), point X(84) www.xtec.es /~qcastell/ttw/ttweng/llistes/l_circumcentre_p2.html   (63 words)

 Contents   (Site not responding. Last check: 2007-10-29) B6 Construct a line perpendicular to a given line from a given point not on the line C5 Construct the tagent to a given circle through a given point on the circle C6 Construct the tagent to a given circle through a given point not on the circle www.dekovsoft.com /info/gc_contents.htm   (421 words)

 symmedian point - OneLook Dictionary Search   (Site not responding. Last check: 2007-10-29) We found 3 dictionaries with English definitions that include the word symmedian point: Tip: Click on the first link on a line below to go directly to a page where "symmedian point" is defined. Symmedian point : A Glossary of Mathematical Terms [home, info] www.onelook.com /?w=symmedian+point   (91 words)

 International Number Federation   (Site not responding. Last check: 2007-10-29) -flv9.euk Note how this is a dynamic proof that the Bottema point M of a vertex A is fixed with respect to the base of a triangle BC and does not depend on the position of A! This is a Also that the the incentre and the Vecton points befriend themselves. In which case the Symmedian point is elevated to the importance of the Circumcentre by Friendship (marriage!) www.fidn.org /flv1.html   (421 words)

 Center for Technology and Teacher Education: Mathematics Activities   (Site not responding. Last check: 2007-10-29) David E. Joyce from the Department of Mathematics and Computer Science at Clark University in Worcester, MA has developed a Geometry applet demonstrating the Euler line of a triangle. Many were developed in the 20th-century, such as: Schiffler Point, Exeter Point, Parry Point, Congruent Isoscelizers Point, Yff Center of Congruence, Isoperimetric Point and Equal Detour Point, Ajima-Malfatti Points, Apollonius Point, Morley Centers, Hofstadter Points, Equal Parallelians Point, Bailey Point. There are also some classical triangle centers constructions: Centroid, Incenter, Circumcenter, Orthocenter, Fermat Point, Nine-point center, Symmedian point, Gergonne point, Nagel point, Mittenpunkt, Spieker center, Feuerbach point, Isodynamic points, and Napoleon points. teacherlink.org /content/math/activities/skp-centerstri2/resources.html   (250 words)

 Symmedian - Encyclopedia Glossary Meaning Explanation Symmedian   (Site not responding. Last check: 2007-10-29) 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)

 International Number Federation   (Site not responding. Last check: 2007-10-29) Proposition: The Symmedian Point, The Orthocentre and The Circumcentre + the 3 vertices of the Triangle the Symmedian Point and the Circumcentre are FRIENDS (on this Hyperbola) Download the Euklides Symmedian point macro (uses the van Lamoen construction). www.fidn.org /itn2.html   (154 words)

 Oops. File not found!   (Site not responding. Last check: 2007-10-29) More recently many of the Geometry pages have been reorganised! Entering at the main Geometry Index Page should allow you to navigate to interesting things about the Miquel Point, Euler Line and so on as well as teaching notes and pages using CabriJava! As always there will be other broken links, if you can be bothered to email me I'd be grateful. s13a.math.aca.mmu.ac.uk /Geometry/.../LemoinePoint/LemoinePtMOrth1.html   (127 words)

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