«Isogonal Conjugates in a Tetrahedron Jawad Sadek, Majid Bani-Yaghoub, and Noah H. Rhee Abstract. The symmedian point of a tetrahedron is deﬁned and ...»
Volume 16 (2016) 43–50.
Isogonal Conjugates in a Tetrahedron
Jawad Sadek, Majid Bani-Yaghoub, and Noah H. Rhee
Abstract. The symmedian point of a tetrahedron is deﬁned and the existence of
the symmedian point of a tetrahedron is proved through a geometrical argument.
It is also shown that the symmedian point and the least squares point of a tetra- hedron are concurrent. We also show that the symmedian point of a tetrahedron coincides with the centroid of the corresponding pedal tetrahedron. Furthermore, the notion of isogonal conjugate to tetrahedra is introduced, with a simple for- mula in barycentric coordinates. In particular, the barycentric coordinates for the symmedian point of a tetrahedron are given.
1. Introduction The symmedian point of a triangle is one of the 6,000 known points associated with the geometry of a triangle . To deﬁne the symmedian point, we begin with the concept of isogonal lines. Two lines AR and AS through the vertex A of an angle are said to be isogonal if they are equally inclined from the sides that form ∠A. The lines that are isogonal to the medians of a triangle are called symme- dian lines , pp. 75-76. Figure 1 (a) shows that the symmedian line AP of the triangle ABC is obtained by reﬂecting the median AM through the corresponding angle bisector AL. The symmedian lines intersect at a single point K known as the symmedian point, also called the Lemoine point. It turns out that the symmedian point of a triangle coincides with the point at which the sum of the squares of the perpendicular distances from the three sides of the triangle is minimum (the least squares point, LSP), . Another property of the symmedian point of a triangle is described below. As shown in Figure 1 (b), let A B C the pedal triangle of K (i.e., the triangle obtained by projecting K onto the sides of the original triangle). Then the symmedian point of the triangle ABC and the centroid of the triangle A B C are concurrent.
The existence of symmedian point of a triangle was proved by the the French mathematician Emile Lemoine in 1873 (, Chapter 7). Later the symmedian point was deﬁned by Marr for equiharmonic tetrahedrons in 1919 . In the present work we provide the deﬁnition and prove the existence of the symmedian point of an arbitrary tetrahedron. Then we show that the symmedian point of a tetrahedron coincides with the LSP of that tetrahedron and the centroid of the corresponding Publication Date: March 13, 2016. Communicating Editor: Paul Yiu.
The authors would like to thank the anonymous reviewer for the valuable comments and sugges- tions to improve the quality of the
petal tetrahedron. Furthermore, we will demonstrate the utility of least squares solution for determining the location of the least squares points and hence the symmedian points.
The rest of this paper is organized as follows. In section 2, the existence of the symmedian point of a tetrahedron is proved. In section 3, it is shown that the symmedian point and LSP of a tetrahedron are concurrent. In section 4, the concurrency of the symmedian point and the centroid of the corresponding petal tetrahedron is proved. In section 5, a discussion of the main results is provided.
2. Symmedian point of a tetrahedron and barycentric coordinates Let ABCD be a tetrahedron. Two planes (P ) and (Q) through AB, for instance, are said to be isogonal conjugates if they are equally inclined from the sides that form the di`dre angle between the planes of the triangles ABC and e ABD. (P ) is called the isogonal conjugate of (Q) and vice versa. If a point X of ABCD is joined to vertex A and vertex B, the plane through XA and XB has an isogonal conjugate at A. Similarly, joining X to vertices B and D, D and C, A and C, B and C, C and D, produce ﬁve more conjugate planes. There is no immediately obvious reason why these six conjugates should be concurrent. However, that this is always the case will follow from lemma 2 below. Let M be the midpoint of CD. The plane containing AB and that is isogonal to the plane of triangle ABM is called a symmedian plane of tetrahedron ABCD. Taking the midpoints of the six sides of the tetrahedron ABCD and forming the associated symmedian planes, we call the intersection point of these symmedian planes the symmedian point of the tetrahedron. In this section we show that all six symmedian planes are indeed concurrent at a point. This deﬁnition of the symmedian point differs from the one given in , which was only deﬁned for equiharmonic tetrahedrons .
For the existence of the symmedian point of an arbitrary tetrahedron, we ﬁrst need the following two lemmas.
Isogonal conjugates in a tetrahedron 45 Lemma 1. All six median planes obtained from a side of a tetrahedron and the midpoint of its opposite side are concurrent.
As shown in Figure 2 (a), let M1 and M2 be the midpoints of the opposite sides CD and AB, respectively. The two median planes constructed from M1 and AB, and from M2 and CD intersect at the line containing the points M1 and M2. Similarly, the other median planes constructed from AC and M3, BD and M4 contain M3 M4, and the planes formed with BC and M5, and AD and M6, contain M5 M6, where M3, M4, M5, and M6, are the midpoints of BD, AC, AD, and BC, respectively. Thus it is enough to show that the line segments M1 M2, M3 M4, and M5 M6 are concurrent. This can be shown by noticing that M1 M4 and M2 M3 are parallel to AD, and M2 M4 and M1 M3 are both parallel to BC. Thus the quadrilateral M1 M3 M2 M4 is a parallelogram. It follows that the diagonals M3 M4 and M1 M2 cross each other at their midpoints. Similar argument shows that the quadrilateral M3 M5 M4 M6 ia a parallelogram with diagonals M5 M6 and M3 M4 crossing each other at their midpoints. The desired result follows.
3. Concurrency of the Symmedian Point and the Least Squares Point The LSP of a given tetrahedron ABCD is the point from which the sum of the squares of the perpendicular distances to the four sides of the tetrahedron ABCD is minimized. Now we show that the symmedian point and the LSP of a tetrahedron are concurrent. We start with the following lemma.
48 J. Sadek, M. Bani Yaghoub and N. H. Rhee
which contradicts (9). So we must have C = K.
Corollary 7. The symmedian point and hence the LSP of a tetrahedron belongs to its interior.
Since K = C and C is in the interior of the petal tetrahedron and the pertal tetrahedron is in the interior of the given tetrahedron, the symmedian point K of the given tetrahedron belongs to its interior.
can be deﬁned as the point of intersection of the lines joining the vertices to the symmedian points of the opposite faces. Now we give an example that shows that our symmedian point is different from Marr’s symmedian point.
Example 1. Consider the tetrahedron ABCD such that A(0, 0, 0), B(1, 0, 0, C(0, 1, 0) and D(0, 0, 1).
Note that the tetrahedron ABCD is equiharmonic and one can ˜ compute Marr’s symmedian point K = (1/5, 1/5, 1/5). Our symmedian point is ˜ = K.
K(1/6, 1/6, 1/6). So K In summary, the merit of the present work is twofold. First, the deﬁnition of the symmedian point of a tetrahedron is a true generalization of the symmedian point of a triangle, because they both coincide with their corresponding least square points.
Second, the notion of isogonal conjugate has been extended to tetrahedra, with a simple formula in barycentric coordinates. In particular, a formula for the symmedian point of a tetrahedron has been given in terms of the barycentric coordinates.
References  A. Bogomolny, All About Symmedians, www.cut-the-knot.org/triangle/symmedians.shtml.
 H. S. M. Coxeter, Introduction to Geometry, Wiley, 1963.
 R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995.
 C. Kimberling, Encyclopedia of Triangle Centers, http://faculty.evansville.edu/ck6/encyclopedia/ETC.html.
 W. L. Marr, The co-symmedian system of tetrahedra inscribed in a sphere, Proceedings of the Edinburgh mathematical society, 37 (1918) 59-64.
 J. Neuberg, Memoire sur la tetraedre, Acad. Royale des Sciences, 37 (1886) 64.
 P. Yiu, The uses of homogeneous barycentric coordinates in plane Euclidean Geometry, Internat.
J. Math. Ed. Sci. Tech., 31 (2000) 569–578.
Jawad Sadek: Department of Mathematics, Computer Science and Information Systems, Northwest Missouri State University, Maryville, Missouri 64468-6001, USA E-mail address: JAWADS@nwmissouri.edu Majid Bani-Yaghoub: Department of Mathematics and Statistics, University of Missouri - Kansas City, Kansas City, Missouri 64110-2499, USA E-mail address: email@example.com
Noah H. Rhee: Department of Mathematics and Statistics, University of Missouri - Kansas City, Kansas City, Missouri 64110-2499, USA E-mail address: RheeN@umkc.edu