Isogons and Isogonic Symmetry


The word isogon has almost completely been replaced by the word equiangular. It describes a polygon with all angles congruent such as rectangles and the regular polygons. The word comes from the Greek isos for same, and gon for knee or corner.
Isogonic is  a related word that describes a type of symmetry between lines, passing through the vertex of an angle, and the angle bisector. In the figure Angle ABC is shown with its bisector BB'. The rays BX and BY are isogonal because they make the same angle with the angle Bisector. 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 isogonal points.  If three lines in a triangle are concurrent, then their isogonic lines are also concurrent.  In the figure the Red segments AA', BB', and CC' intersect at Point X.  The three blue rays are the isogonic lines for the three Red Segments, which are reflected about the angle bisectors (dashed rays).  Blue Rays intersect in a single point also, labled X'.  Points X and X' are called isogonal conjugates.

One famous pair of isogonal conjugates is the orthocenter (intersection of the altitudes) and the circumcenter (center of the circle which circumscribes a triangle). If you draw any triangle and find these two points (lets call them P and Q), then draw the angle bisector from any vertex of the triangle (which we will call AX, you will see that the angles PAX and QAX are congruent.

It is well known that the pedal triangle of the orthocenter has all three vertices laying on the famous Nine-point circle. The "feet" of the circumcenter of ABC are the midpoints of the sides a, b, and c, and so the pedal triangle of the circumcenter also has vertices laying on the nine point circle. Even more enchanting is that the center of the nine point circle is the center of the line segment joining the orthocenter and the circumcenter. 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.
 
 

  Some isogonal reflections have special names.  If we reflect the medians of a triangle in the angle bisectors, the reflections are called symmedians, a contraction of the term "symmetric medians".  The symmedians intersect in a point called the Lemoine Point, for French engineer and mathematician Emile Lemoine.  Lemoine's short biography can be found at the St Andrews University web site. In Germany the point is often called Grebe's point. You can find a brief history of his life and some referrences below. In the figure, the medians are in red, the angle bisectors are dashed, and the symmedians are in blue.  The Lemoine point is point L (historically it should be k, see history notes below).   If we call X the point where the symmedian from A intersects side BC, then the ratio of BX / XC = AB2 / AC2. Thus in a triangle ABC, the three points on side c cut by the median, the angle bisector, and the symmedian respectively, allow us to divide side c of the triangle into ratios of 1:1, a:b, or a2:b2.
  The Lemoine Circle is defined by three lines through the Lemoine point parallel to the sides of the triangle.  These lines intersect the triangle in six points, which all lie on the Lemoine Circle.  The center of the Lemoine Circle is the midpoint of a line segment from the Lemoine point to the circumcenter. I have copied a post from the Historia Matematica discussion group below which gives some of the history of the point and its name.


The symmedians are related to another triangle center called the Gergonne point, named for Joseph Gergonne (1771-1859).  A very short biography is at St Andrews University web site.  The Gergonne point (G' in figure below right) is the point of intersection of the lines (red) from the vertices of a triangle to the points where the inscribed circle (dashed line) is tangent to the triangle (A', B', and C').  The triangle (blue) connecting the three points of tangency is called the Gergonne Triangle or the Contact Triangle.  If you construct the symmedians of the Gergonne Triangle, you will observe that the isogonic reflection of the medians is the lines (red) from the vertices to the points of tangency. This means the Lemoine Point of the Gergonne Triangle is the Gergonne Point of the original triangle, ABC.

Here is a link where you can find a proof of Gergonne point concurrence using Ceva's Theorem.

The Gergone point is the intersection of the segments from a vertex to the tangent point of the incircle with the opposite side. If instead, we construct the three excircles, and find the segments joining each vertex of the triangle to the tangent point of the opposite excircle, the segments are concurrent at a point called the Nagel Point. The Nagel point, named for German mathematician Christian Heinrich von Nagel, lies on a line that also contains the in-center and the centroid of the triangle. The line is often called the Nagel Line. A brief biography of Nagel can be found here The illustration is from the excellent geometry site created by Antonio Gutierrez.

If the antiparallels to the three sides passing through the Lemoine point are extended to the six points where they intersect the three sides, another circle will pass through these points. 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..


Ernst Wilhelm Grebe Because I could find no links to a History of Grebe on the net, I have copied a post from Julio Gonzalez Cabillon to the Historia Matematica discussion group which includes information about his life and some referrences about his work.
For what it is worth, let me say that Ernst Wilhelm Grebe was a German teacher of mathematics [more precisely _Oberlehrer am Gym- nasium_] at Kassel, and was born on August 30, 1804 -- same birth date (Aug. 30) for Joseph Serret, Carle Runge, Olga Taussky-Todd, among others.
Ernst Grebe is remembered only for a thoughtful paper appeared in 1847 [2] concerning some interesting properties of the triangle:
If on each side of a given (arbitrary) triangle ABC one describes a square ( exterior to ABC ), then the extended outside sides of the squares, thus obtained, form a similar triangle A'B'C'. The center of similarity of both triangles is the meeting point of the straight lines AA', BB', CC'. In German this point was first called _Grebe'schen Punkt_ [Grebe's point], a TERM which seems to have been first referred to by E. Hain as early as 1875, in his paper "Ueber den Grebe'schen Punkt" [ _Archiv der Mathematik und Physik_ (= Grunert's _Archive_) volume LVIII (1876), pp. 84-89 ]. Afterwards, the term _Grebe'schen Punkt_ appeared many times in the _Jahrbuch ueber die Fortschritte der Mathematik_ by reviewers such as Dr. Schemmel (Berlin, 1875), Prof. Mansion (Gent, 1881), Prof. Lampe (Berlin, 1881), Dr. Lange (Berlin, 1885), et cetera.
The first math contribution of Grebe that I am aware of is a book titled _De Linea Helice_ [1], in Latin, and published in Marburg. As you may read below, Grebe wrote many articles and books mainly on Geometry.
Grebe died in Kassel, on January 14, 1874 (same date, January 14, for Nicolaus Mercator, Edmond Halley, George Berkeley, Charles Bossut, Charles Dodgson (= Lewis Carroll), Charles Hermite, Kurt Go"del, William Feller, among many others).
Grebe's books and papers:
[1] De linea helice ejusque projectionibus orthographicis commen- tatio, quam ad summos in philosophia honores rite Capessendos Amplissimo Philosophorum Marburgensium Ordini. 59 pages. Marburg, 1829.
[2] Das geradlinige Dreieck in Bezug auf die Quadrate der Per- pendikel, die man von einem Punkte seiner Ebene auf seine Seiten faellen kann, 9 pages, Grunert's _Archiv_ 9 (1847).
[3] Ueber die Verwandlung der Wurzeln quadratischer Gleichungen in Kettenbrueche, Cassel, 48 pages, 1847.
[4] Aufloesung reiner Gleichungen, insbesond solcher des 3. Gra- des durch Kettenbrueche 99 pages, Grunert's _Archiv_ 10 (1847) & 16 (1851).
[5] Eroerterung einer Spielerei durch d. Wahrscheinlichtkeit- Rechnung, 2 pages, Grunert's _Archiv_ 11 (1848).
[6] Beweis einer Formel fuer \pi, 6 pages, Grunert's _Archiv_ 12 (1849).
[7] Rationalmachen von Nennern mit unbestimmt vielen irrationalen Gliedern, 5 pages, Grunert's _Archiv_ 13 (1849).
[8] Theilung eines ebenen Dreiecks durch 2 sich innerhalb dessel- ben schneidende Geraden in 4 gleiche Flaechenstuecke, 3 pages, Grunert's _Archiv_ 13 (1849).
[9] Ausdruecke, welche fuer Wurzeln hoeheren Grades mit (B+ A sqra)(B - A sqr a) analog sind, 5 pages, Grunert's _Archiv_ 13 (1849).
[10] Ueber das Auffinden von Dreiecken, deren Seiten sich gleich- zeitig mit den Halbierungs-Linien durch ganze Zahlen ausdruecke lassen, 11 pages, Grunert's _Archiv_ 17 (1851).
[11] Vergleich zwischen dem arithmetischen, geometrischen & har- monischen Mittel, Schloemilch Journal, 1/2 page, 3 (1858).
[12] Das prismatoid, 4 pages, Grunert's _Archiv_ 39 (1862).
[13] Formeln der sphaerischen Trigonometrie, 3 pages, Grunert's _Archiv_ 39 (1862).
[14] Beitraege zur Lehre von dem geradlinigen Dreieck, Cassel: Doell & Schaefer, 16 pages, 1862.
[15] Ueber einen Satz der Geometrie (Jubilaeumsschrift des Dr Gerling), Kassel, 1862.
[16] Lehrsatz der Geometrie, 2 pages, Schloemilch's Journal 8 (1863).
[17] Zusammenstellung von Stuecken rationaler ebener Dreiecke, Halle: Schmidt, 248 pages, 1864.
[18] Fuenfzig Aufgaben ueber das geradlinige Dreieck trigonome- trisch geloest, Cassel: Doell & Schaefer, 13 pages, 1865.
Re: "Also, why is it symbolized with K?"
At the time, the choice of "K" came as the first 'unused' letter of the alphabet in Lemoine's scheme, since the first letters A, B, C, ... were (and still are) used for naming the vertices of a triangle, and the letters F, G, H, I had already a standard and somehow accepted meaning:
F: [F]euerbach point (= center of the 9-point circle) G: [G]ravity center (= Barycenter = Median point = Centroid) H: [H]oehen(schnitt) punkt (= Orthocenter) I: [I]ncenter
Therefore, within Lemoine's scheme, there was:
J: used for other purposes (especially, J_a, J_b, & J_c) K: available


Some History notes related to Lemoine Point Here is a quote from a post by Antreas P. Hatzipolakis with some clips from Professor Clark Kimberling's web page. .

In Clark Kimberling's web page for Emile Michel Hyacinthe Lemoine we read:

{quote} Many of Lemoine's contributions can be found in Nathan Altshiller Court, College Geometry. Second edition. Barnes & Noble, New York, 1969.
In addition to Lemoinian Geometry (14 pages in Chapter 10), Court presents a section of Historical and Bibliographical Notes, in which he writes:
In 1873, Lemoine read a paper before the Association Franc,aise pour l'Avancement des Sciences [meeting in Lyon] entitled "Sur quelques proprietes d'un point remarquable du triangle." The paper appeared in the Proceedings pp. 90-91, and was also published in [Nouvelles annales de mathematiques] ... The paper may be said to have laid the foundations not only of Lemoinian Geometry, but also of the modern geometry of the triangle as a whole.
Court goes on to explain that the basic point of Lemoinian geometry was called by Lemoine the center of antiparallel medians. In 1884, J. Neuberg gave it the name Lemoine point. However, the point had been earlier noted by LHuilier in 1809, Grebe in 1847, and others prior to Lemoine's extensive discussion in 1873. For awhile, the point was called the Lemoine point in France and the Grebe point in Germany. In 1883, M. d'Ocagne introduced the term symmedian for a line obtained as the reflection of a median of a triangle about the corresponding angle bisector (i.e., Lemoine's "antiparallel median". 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 . This name is now widely accepted.