Let P and Q be distinct points in a plane, and let T be a triangle rigidly rotating about P. For a list of other harmonic conjugates of X 6 , click Tables at the top of this page. If you have The Geometer's Sketchpad, you can view Gergonne point. If you have GeoGebra, you can view Gergonne point.
X 7 lies on the Lucas cubic and these lines: 1,20 2,9 3, 4, 6, 8,65 11, 12, 21,56 27,81 37, 33, 34, 55, 58, 59, 72, 73, 76, 80, 92, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , For a list of other harmonic conjugates of X 7 , click Tables at the top of this page. Also, X 8 is the incenter of the anticomplementary triangle.
X 8 is the radical center of Oa, Ob, Oc. Randy Hutson, April 9, Randy Hutson, September 14, Let A 28 B 28 C 28 be the Gemini triangle Another construction is given by Xavier Dussau: Elementary construction of the Nagel point. April 29, If you have The Geometer's Sketchpad, you can view Nagel point. If you have GeoGebra, you can view Nagel point. For a list of other harmonic conjugates of X 8 , click Tables at the top of this page.
Let A' be the orthocorrespondent of the A-excenter, and define B' and C' cyclically. Let A'B'C' be the orthic triangle. Let E be the locus of the trilinear pole of a line that passes through X 1.
The center of E is X 9. Also, E is a circumellipse of ABC and an inellipse of the excentral triangle. The locus E also passes through the vertices of Gemini triangle 2. Let A' be the intersection of the tangents to the A-excircle at the intercepts with the circumcircle, and define B' and C' cyclically. Randy Hutson, December 2, Let A' be the perspector of the A-mixtilinear excircle, and define B' and C' cyclically.
Then ABC is a billiard orbit of E 3-periodic. If we fix E in the plane, all its triangular orbits a set of "rotating" triangles T have the same X 9. Note that X 9 is the point of concurrence of lines drawn from each excenter to the midpoint of the corresponding side of T. An online pdf is available: Click here.
E passes through the point X i for these i: 88, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , The perspector of E is X 1 ; the major axis of E passes through X i for these i: 9, , , , and the minor axis, for these i: 9, , , The ellipse E is the isogonal conjugate of the antiorthic axis.
The axes of E are the asymptotes of the Feuerbach hyperbola. The triangle tangent to the vertices of E is the excentral triangle. The ellipse E passes through A', B', C'. See Three orbits in elliptic billiard. Dan Reznik, July 1, The circumellipse with center X 9 meets the circumhyperbola with center X 11 i.
Dan Reznik, January 20, Each of the following cubics passes through the four foci two real and two imaginary of the ellipse E: K, K, K, K, K, K The two real foci are a pair of isogonal conjugates, and likewise for the two imaginary foci.
Peter Moses, July 2 and 3, This line, denoted by L 31 , is the perspectrix of the anticomplementary triangle and the inner Conway triangle which is the intouch triangle of the anticomplementary triangle.
For a dynamic graphic with many options, based on a triangle inscribed in the circumellipse with center X 9 , see Elliptic Billiard: Loci of Triangular Centers.
Dan Reznik, February 4, If you have The Geometer's Sketchpad, you can view Mittenpunkt. If you have GeoGebra, you can view Mittenpunkt.
X 9 lies on the these conics: Feuerbach circumhyperbola, Feuerbach circumhyperbola of the medial triangle, Jerabek circumhyperbola of the excentral triangle, Mandart hyperbola. Note that A'B'C' is the complement of the excentral triangle, and the extraversion triangle of X The Spieker circle is the incircle of the medial triangle; its center, X 10 , is the centroid of the perimeter of ABC.
If you have The Geometer's Sketchpad, you can view Spieker center. If you have GeoGebra, you can view Spieker center. A construction of X 10 is given at Antreas Hatzipolakis, August 29, Let A'B'C' be the excentral triangle. Randy Hutson, July 31 Let A 20 B 20 C 20 be the Gemini triangle X 10 lies on the Kiepert hyperbola and these lines: 1,2 3, 4,9 5, 6, 11, 12,65 20, 21,35 28, 29, 31, 33, 34, 36, 37, 38, 39, 44, 46,63 55, 56, 57, 58, 69, 75,76 81, 82,83 86, 87, 92, 98, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , For a list of other harmonic conjugates of X 10 , click Tables at the top of this page.
X 10 is the internal center of similitude of the Apollonius and nine-points circles. Let A'B'C' be the 2nd extouch triangle. Also, let A''B''C'' be the 1st circumperp triangle. X 11 is the point of tangency of the nine-point circle and the incircle. The nine-point circle is the circumcenter of the medial triangle, as well as the orthic triangle.
Feuerbach's famous theorem states that the nine-point circle is tangent to the incircle and the three excircles. Randy Hutson, March 25, The circumcircle of the incentral triangle intersects the incircle at 2 points, X 11 and X , and the nine-point circle at 2 points, X 11 and X If you have The Geometer's Sketchpad, you can view Feuerbach point.
If you have GeoGebra, you can view Feuerbach point. For a list of other harmonic conjugates of X 11 , click Tables at the top of this page. X 11 lies on the incentral circle, Mandart circle, cevian circle of every point on the Feuerbach hyperbola, and these lines: 1,5 2,55 3, 4,56 7, 8, 9, 10, 13, 14, 28, 30,36 33, 34, 35, 57, 65, 68, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , X 12 lies on these lines: 1,5 2,56 3, 4,55 7, 10,65 17, 18, 30,35 33, 34, 36, 37, 38, 40, 42, 54, 57, 63, 71, 79, 85, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , For a list of other harmonic conjugates of X 12 , click Tables at the top of this page.
If you have The Geometer's Sketchpad, you can view X If you have GeoGebra, you can view X The antipedal triangle of X 13 is equilateral. To represent the Fermat point in the form f a,b,c : f b,c,a : f c,a,b , one must use Boolean variables, as shown at Fermat point. X , center of the Evans Conic and 19 other triangle centers.
If you have GeoGebra, you can view 1st isogonic center. The Evans conic is introduced in Evans, Lawrence S. Randy Hutson, January 29, Paul Hanna and Peter Moses, August 6, Kiminari Shinagawa, February 20, Dao Thanh Oai, The points A 0 , B 0 , C 0 are collinear. Dan Reznik, August 26, Peter Moses, April 22, Several notable circles pass through X 13 and X For each circle listed here, the appearance of i; [name], [list] means that the center is X i , and the points with listed indices lie on the circle:.
X 13 lies on the Neuberg cubic and these lines: 2,16 3,17 4,61 5,18 6,14 11, 15,30 76, 80, 98, 99, , , , , , , , , , , For a list of other harmonic conjugates of X 13 , click Tables at the top of this page.
The antipedal triangle of X 14 is equilateral. If you have The Geometer's Sketchpad, you can view 2nd isogonic center If you have GeoGebra, you can view 2nd isogonic center. X 14 lies on the Neuberg cubic and these lines: 2,15 3,18 4,62 5,17 6,13 11, 16,30 76, 98, 99, , , , , , , , , , , , For a list of other harmonic conjugates of X 14 , click Tables at the top of this page. The circle having diameter UV is the A-Apollonian circle.
The B- and C- Apollonian circles are similarly constructed. Each circle passes through one vertex and both isodynamic points. The pedal triangle of X 15 is equilateral. Several notable circles pass through X 15 and X This list of circles was contributed by Peter Moses, April 17, , with the following notes. Each circle passes through a vertex and both isodynamic points. The pedal triangle of X 16 is equilateral.
If you have The Geometer's Sketchpad, you can view 2nd isodynamic point. If you have GeoGebra, you can view 2nd isodynamic point. Let X,Y,Z be the centers of the equilateral triangles in the construction of X If you have The Geometer's Sketchpad, you can view 1st Napoleon point. If you have GeoGebra, you can view 1st Napoleon point. X 17 lies on the Napoleon cubic and these lines: 2,62 3,13 4,15 5,14 6,18 12, 16, 76, 83, , , , , For a list of other harmonic conjugates of X 17 , click Tables at the top of this page.
If you have The Geometer's Sketchpad, you can view 2nd Napoleon point. If you have GeoGebra, you can view 2nd Napoleon point. X 18 lies on the Napoleon cubic and these lines: 2,61 3,14 4,16 5,13 6,17 12, 15, 76, 83, , , , , For a list of other harmonic conjugates of X 18 , click Tables at the top of this page. X 19 is the homothetic center of the orthic and extangents triangles.
If you have The Geometer's Sketchpad, you can view Clawson point. If you have GeoGebra, you can view Clawson point. Further information is available from Paul Yiu's Website. Although John Clawson studied this point in , it was studied earlier by Lemoine:. This article is available online at Numdam. Let A'B'C' be the 4th Brocard triangle. Let A" be the trilinear product of the real or imaginary circumcircle intercepts of line B'C'. Randy Hutson, December 26, Let A'B'C' be the hexyl triangle.
Let La be the trilinear polar of A', and define Lb and Lc cyclically. Let A" be the intersection of the tangents to the ellipse at Ba and Ca, and define B" and C" cyclically. Note: X 4 is the perspector of any circle centered at a vertex of ABC. Triangle A"B"C" is homothetic to the anticomplementary triangle, and the center of homothety is X 20 , which is also the orthocenter of A"B"C".
Then La,Lb,Lc concur in X Also, let A'B'C' be the cevian triangle of X Let L be the Brocard axis of the intouch triangle. If you have GeoGebra, you can view De Longchamps point. The name of this point honors Kurt Schiffler. This configuration extends to Kirikami-Schiffler points and generalizations found by Peter Moses, as introduced just before X Let A'B'C' be the 2nd circumperp triangle.
Let La be the tangent to the nine-point circle at A', and define Lb and Lc cyclically. Let A" be the isogonal conjugate of the trilinear pole of La, and define B" and C" cyclically.
Then X 21 is the radical center of Oa, Ob, Oc. If you have The Geometer's Sketchpad, you can view Schiffler point. If you have GeoGebra, you can view Schiffler point. For a list of other harmonic conjugates of X 21 , click Tables at the top of this page. See the note just before X for a generalization. The triangle A'B'C' is homothetic to the anticomplementary triangle, and the center of homothety is X Randy Hutson, September 5, If you have The Geometer's Sketchpad, you can view Exeter point.
If you have GeoGebra, you can view Exeter point. For a list of other harmonic conjugates of X 22 , click Tables at the top of this page. Let A'B'C' be the antipedal triangle of X 3 the tangential triangle.
Randy Hutson, Octobe3r 13, Let A'B'C' be the anti-orthocentroidal triangle. If you have The Geometer's Sketchpad, you can view Far-out point. If you have GeoGebra, you can view Far-out point.
For a list of other harmonic conjugates of X 23 , click Tables at the top of this page. For a list of other harmonic conjugates of X 24 , click Tables at the top of this page. Constructed as indicated by the name; also X 25 is the pole of the orthic axis the line having trilinear coefficients cos A : cos B : cos C with respect to the circumcircle. Let A' be the radical center of the nine-point circle and the B- and C-power circles. The triangle A'B'C' is homothetic with the orthic triangle, and the center of homothety is X Also X 25 is the point of intersection of these two lines: isotomic conjugate of polar conjugate of van Aubel line i.
Also, X 25 is the trilinear pole of line X X , this line being the isogonal conjugate of the isotomic conjugate of the orthic axis; the line X X is also the polar of X 76 wrt polar circle, and the line is also the radical axis of circumcircle and 2nd Lemoine circle.
Let A" be the barycentric product of the real or imaginary circumcircle intercepts of line B'C'. Define B", C" cyclically. Randy Hutson, October 27, The 2nd Ehrmann triangle, defined in the preamble to X , can be generalized as follows.
Define Ac symmetrically, and define Ba and Cb cyclically. The X 1 -Ehrmann triangle is the intangents triangle, and the X 6 -Ehrmann triangle is the 2nd Ehrmann triangle. If P lies on the circumcircle, the P-Ehrmann triangle is the tangential triangle.
The homothetic center of the orthic triangle and the X 4 -Ehrmann triangle is X Let H X denote hodpoint of a point X, as defined in the preamble just before X Dasari Naga Vijay Krishna, March 14, For a list of other harmonic conjugates of X 25 , click Tables at the top of this page. Theorems involving X 26 , published in by A.
Gob, are discussed in Roger A. Johnson, Advanced Euclidean Geometry, Dover, , For a list of other harmonic conjugates of X 26 , click Tables at the top of this page. For a list of other harmonic conjugates of X 27 , click Tables at the top of this page. For a list of other harmonic conjugates of X 28 , click Tables at the top of this page. If you have GeoGebra, you can view X 29 and X 29 cevapoint.
X 29 lies on these lines: 1,92 2,3 8, 10, 33,78 34,77 58, 65, 81, , , , , , , , , , , , , , , , , , , , , , , , For a list of other harmonic conjugates of X 29 , click Tables at the top of this page. X 30 is the point of intersection of the Euler line and the line at infinity. Thus, each of the lines listed below is parallel to the Euler line. Let A'B'C' be the circumsymmedial triangle. Let A'B'C' be the Apus triangle.
Let A'B'C' be the Ara triangle. Define the 1st Kenmotu diagonals triangle as the triangle formed by the diagonals of the squares in the Kenmotu configuration with center X that do not include X Define the 2nd Kenmotu diagonals triangle as the triangle formed by the diagonals of the squares in the Kenmotu configuration with center X that do not include X X 31 is the Brianchon point perspector of the inellipse that is the trilinear square of the Lemoine axis.
The center of the inellipse is X Randy Hutson, October 15, For a list of other harmonic conjugates of X 31 , click Tables at the top of this page. The 5th Brocard triangle is here introduced as the vertex triangle of the circumcevian triangles of the 1st and 2nd Brocard points. The 5th Brocard triangle is homothetic to ABC at X 32 , homothetic to the medial triangle at X , homothetic to the anticomplementary triangle at X , perspective to the 1st Brocard triangle at X , and perspective to the 3rd Brocard triangle at X Let A'B'C' be the 1st Brocard triangle.
X 32 is the Brianchon point perspector of the inellipse that is the barycentric square of the Lemoine axis. The center of this inellipse is X X 32 lies on these lines: 1, 2,83 3,6 4,98 5, 9, 20, 21, 22, 24, 25, 31,41 35, 48, 51, 55, 56, 71, 75, 76, 81, 99, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Do they all meet at one point? Can you balance the triangle at that point?
Try this: drag the points above until you get a right triangle just by eye is OK. Where is the circumcenter? Try this: find the incenter of a triangle using a compass and straightedge at: Inscribe a Circle in a Triangle. Wolfram Alpha » Explore anything with the first computational knowledge engine. Wolfram Demonstrations Project » Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.
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