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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, symmedians are three particular lines associated with every
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
. They are constructed by taking a
median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
of the triangle (a line connecting a
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...
with the midpoint of the opposite side), and reflecting the line over the corresponding
angle bisector In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
(the line through the same vertex that divides the angle there in half). The angle formed by the symmedian and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is on the other side of the angle bisector. The three symmedians meet at a
triangle center In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For exampl ...
called the Lemoine point. Ross Honsberger has called its existence "one of the crown jewels of modern geometry"..


Isogonality

Many times in geometry, if we take three special lines through the vertices of a triangle, or '' cevians'', then their reflections about the corresponding angle bisectors, called ''isogonal lines'', will also have interesting properties. For instance, if three cevians of a triangle intersect at a point , then their isogonal lines also intersect at a point, called the
isogonal conjugate __notoc__ In geometry, the isogonal conjugate of a point with respect to a triangle is constructed by reflecting the lines about the angle bisectors of respectively. These three reflected lines concur at the isogonal conjugate of . (Th ...
of . The symmedians illustrate this fact. * In the diagram, the medians (in black) intersect at the
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
. * Because the symmedians (in red) are isogonal to the medians, the symmedians also intersect at a single point, . This point is called the triangle's symmedian point, or alternatively the Lemoine point or Grebe point. The dotted lines are the angle bisectors; the symmedians and medians are symmetric about the angle bisectors (hence the name "symmedian.")


Construction of the symmedian

Let be a triangle. Construct a point by intersecting the
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
s from and to the
circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
. Then is the symmedian of . ''first proof.'' Let the reflection of across the angle bisector of meet at . Then: \frac = \frac =\frac\frac =\frac\frac =\frac\frac=1 ''second proof.'' Define as the
isogonal conjugate __notoc__ In geometry, the isogonal conjugate of a point with respect to a triangle is constructed by reflecting the lines about the angle bisectors of respectively. These three reflected lines concur at the isogonal conjugate of . (Th ...
of . It is easy to see that the reflection of about the bisector is the line through parallel to . The same is true for , and so, is a parallelogram. is clearly the median, because a parallelogram's diagonals bisect each other, and is its reflection about the bisector. ''third proof.'' Let be the circle with center passing through and , and let be the
circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
of . Say lines intersect at , respectively. Since , triangles and are similar. Since :\angle PBQ = \angle BQC + \angle BAC = \frac = 90^\circ, we see that is a diameter of and hence passes through . Let be the midpoint of . Since is the midpoint of , the similarity implies that , from which the result follows. ''fourth proof.'' Let be the midpoint of the arc . , so is the angle bisector of . Let be the midpoint of , and It follows that is the
Inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
of with respect to the circumcircle. From that, we know that the circumcircle is an Apollonian circle with foci . So is the bisector of angle , and we have achieved our wanted result.


Tetrahedra

The concept of a symmedian point extends to (irregular) tetrahedra. Given a tetrahedron two planes through are isogonal conjugates if they form equal angles with the planes and . Let be the midpoint of the side . The plane containing the side that is isogonal to the plane is called a symmedian plane of the tetrahedron. The symmedian planes can be shown to intersect at a point, the symmedian point. This is also the point that minimizes the squared distance from the faces of the tetrahedron..


References

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External links


Symmedian and Antiparallel
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

Symmedian and 2 Antiparallels
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

Symmedian and the Tangents
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

An interactive Java applet for the symmedian point


Straight lines defined for a triangle