Symmedian
In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the corresponding angle bisector (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 called the Lemoine point. Ross Honsberger has called its existence "one of the crown jewels of modern geometry".<ref name="h">Template:Citation.</ref>
IsogonalityEdit
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 Template:Mvar, then their isogonal lines also intersect at a point, called the isogonal conjugate of Template:Mvar.
The symmedians illustrate this fact.
- In the diagram, the medians (in black) intersect at the centroid Template:Mvar.
- Because the symmedians (in red) are isogonal to the medians, the symmedians also intersect at a single point, Template:Mvar.
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 symmedianEdit
Let Template:Math be a triangle. Construct a point Template:Mvar by intersecting the tangents from Template:Mvar and Template:Mvar to the circumcircle. Then Template:Mvar is the symmedian of Template:Math.<ref>Template:Cite book</ref>
First proof. Let the reflection of Template:Mvar across the angle bisector of Template:Math meet Template:Mvar at Template:Mvar. Then:
<math>\frac{|BM'|}{|M'C|} = \frac{|AM'|\frac{\sin\angle{BAM'}}{\sin\angle{ABM'}}}{|AM'|\frac{\sin\angle{CAM'}}{\sin\angle{ACM'}}} =\frac{\sin\angle{BAM'}}{\sin\angle{ACD}}\frac{\sin\angle{ABD}}{\sin\angle{CAM'}} =\frac{\sin\angle{CAD}}{\sin\angle{ACD}}\frac{\sin\angle{ABD}}{\sin\angle{BAD}} =\frac{|CD|}{|AD|}\frac{|AD|}{|BD|}=1</math>
Second proof. Define Template:Mvar as the isogonal conjugate of Template:Mvar. It is easy to see that the reflection of Template:Mvar about the bisector is the line through Template:Mvar parallel to Template:Mvar. The same is true for Template:Mvar, and so, Template:Mvar is a parallelogram. Template:Mvar is clearly the median, because a parallelogram's diagonals bisect each other, and Template:Mvar is its reflection about the bisector.
Third proof. Let Template:Mvar be the circle with center Template:Mvar passing through Template:Mvar and Template:Mvar, and let Template:Mvar be the circumcenter of Template:Math. Say lines Template:Mvar intersect Template:Mvar at Template:Mvar, respectively. Since Template:Math, triangles Template:Math and Template:Math are similar. Since
- <math>\angle PBQ = \angle BQC + \angle BAC = \frac{\angle BDC + \angle BOC}{2} = 90^\circ,</math>
we see that Template:Mvar is a diameter of Template:Mvar and hence passes through Template:Mvar. Let Template:Mvar be the midpoint of Template:Mvar. Since Template:Mvar is the midpoint of Template:Mvar, the similarity implies that Template:Math, from which the result follows.
Fourth proof. Let Template:Mvar be the midpoint of the arc Template:Mvar. Template:Mvar, so Template:Mvar is the angle bisector of Template:Math. Let Template:Mvar be the midpoint of Template:Mvar, and It follows that Template:Mvar is the Inverse of Template:Mvar with respect to the circumcircle. From that, we know that the circumcircle is an Apollonian circle with foci Template:Mvar. So Template:Mvar is the bisector of angle Template:Math, and we have achieved our wanted result.
TetrahedraEdit
The concept of a symmedian point extends to (irregular) tetrahedra. Given a tetrahedron Template:Mvar two planes Template:Mvar through Template:Mvar are isogonal conjugates if they form equal angles with the planes Template:Mvar and Template:Mvar. Let Template:Mvar be the midpoint of the side Template:Mvar. The plane containing the side Template:Mvar that is isogonal to the plane Template:Mvar 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.<ref name="SBR">Template:Citation.</ref>