Does a Tetrahedron Have an Eüler Line?

Does a TetrahedronHave an Eüler Line? PCMI Summer 2011 Park City, Utah Thursday, July 6th, 2011 Troy Jones Westlake High School, Saratoga Springs, Utah tjones@alpinedistrict.org

Important Vocabulary • Concur lines/planes • Coincide points/lines/planes • Collinear points • Cyclic points • Coplanar points • Cospherical points

“Math Curse”, Jon Scieszka, Viking Press, 1995

Modern Pure Solid Geometry by Nathan Altshiller-Court The Macmillan Company 1935 • Geometric Constructions in Space When Speaking of constructions in space we assume that we are able • to construct a plane given three of its points not lying in a straight line; • to construct the line of intersection of two planes; • to construct the point of intersection of a line and a plane; • to carry out all plane constructions in any plane. These constructions are purely theoretical, for we have no practical method to actually carry them out.

What is the locus of balancing points for these thin strips of paper in the approximate shape of a triangle?

The locus of balancing points will form a segment connecting a vertex with the midpoint of the opposite side. This segment is called a median.

Slicing the triangle into strips parallel to any side, and tracing the midpoints, will create three different balancing lines, or medians.

The medians will concur at a point called the centroid, which will be the balancing point, or center of mass, for the triangle.

Question: How can we extend this idea of finding the center of mass of a triangle in two-dimensions, to finding the center of mass of a tetrahedron in three-dimensions?

We could slice a tetrahedron into thin slices parallel to a face. Each slice would be a triangle that balances at its centroid.

The locus of these centroids would form a segment connecting a vertex of the tetrahedron with the centroid of the face opposite the vertex.

This segment in the tetrahedron is the analog of the median in a triangle. It is the locus of the centroids of each triangular slice, and a skewer piercing the tetrahedron along this segment will balance it.

This segment is referred to as the median of a tetrahedron. The four medians concur at the centroid of the tetrahedron, which is its center of mass, or balancing point.

Another way to construct the centroid of a tetrahedron is to slice the tetrahedron with a plane parallel to a pair of opposite edges. This cross section forms a parallelogram. The parallelogram balances at its centroid.

The locus of the centroid of the parallelogram is the midsegment of the tetrahedron, connecting the midpoints of opposite edges of the tetrahedron.

The three midsegments of a tetrahedron also concur at the centroid of the tetrahedron. The centroid is the midpoint of each midsegment.

The four medians and the three midsegments all concur at the centroid of the tetrahedron.

In a triangle, the centroid divides each median into two segments that are in the ratio 1:2. Another way to state this; the ratio of the shorter segment to the entire median is 1:3.

In a tetrahedron, the centroid divides each median into two segments that are in the ratio 1:3. Another way to state this; the ratio of the shorter segment to the entire median is 1:4.

The locus of points in a plane equidistant from the endpoints of a segment form a line called the perpendicular bisector of the segment.

The three perpendicular bisectors of a triangle concur at a point that is equidistant from all three vertices of the triangle. This point is called the circumcenter, the center of the circumscribed circle.

The locus of points in space equidistant from the endpoints of a segment form a plane called the perpendicular bisector plane, or perpendicular bisector,of the segment.

The perpendicular bisectors of the three edges of a face intersect in a line that is perpendicular to that face through the circumcenter of that face.

Here is just the line of intersection of the perpendicular bisectors of the edges of facePQR.

The perpendicular bisectors of the six edges of a tetrahedron concur at a point called the circumcenter of the tetrahedron.

The circumcenter of the tetrahedron is equidistant from all four vertices, and is the center of the circumscribed sphere of the tetrahedron.

In a triangle, the segments through a vertex, perpendicular to the opposite sides, are called altitudes. The three altitudes of a triangle concur at the orthocenter. The three vertices and the orthocenter form an orthocentric set of points.

In general, the altitudes of a tetrahedron do not concur. (Not every tetrahedron has an orthocenter).

If the altitudes do concur, the point of concurrency is called the orthocenter of the tetrahedron, and the tetrahedron is called orthocentric.

In a triangle, the centroid, circumcenter, and orthocenter are collinear. This line is called the Eüler line. The centroid divides the Eüler segment into a 1:2 ratio.

Question: Does a tetrahedron have an Eüler line?

In an orthocentric tetrahedron, the centroid, circumcenter, and orthocenter are collinear, and this line may, by analogy with the plane, be called the Eüler line of the tetrahedron. The centroid is the midpoint of the Eüler segment.

If the tetrahedron is not orthocentric, the orthocenter does not exist. But there is a point collinear with the circumcenter and centroid with properties similar to the orthocenter. Construct the plane through the midpoint of an edge and perpendicular to the opposite edge.

The six planes through the midpoints of each edge and perpendicular to each corresponding opposite edge concur at the Monge point. The Monge point is collinear with the circumcenter and the centroid. The centroid is the midpoint of the segment between the Monge point and the orthocenter.

In an orthocentric tetrahedron, the Monge point and the orthocenter coincide. GaspardMonge 1746-1818

The triangle formed by connecting the midpoints of the sides of a triangle is called the medial triangle. The circumcenter of the medial triangle is the midpoint of the Eüler segment.

The circumcircle of the medial triangle is usually referred to as the nine-point circle. It contains the 3 midpoints of the sides, the 3 feet of the altitudes, and the 3 midpoints of the segments connecting the orthocenter with each vertex.

The tetrahedron formed by connecting the centroids of the faces of a tetrahedron is called the medial tetrahedron. The circumcenter of the medial tetrahedron circumsphereis on the Eüler segment, and it divides the Eüler segment in a 1:2 ratio.

The circumsphere of the medial tetrahedron is called the twelve-point sphere. It contains the 4 centroids of the faces, 4 points that are 1/3 the way from the Monge point to each vertex, and the 4 feet of the perpendicular line from each 1/3 point to the opposite face.

Given an orthocentric set of 4 points in a plane, the nine-point circle is the same, no matter which 3 points you choose as the vertices of the triangle.

Every face of a tetrahedron has a nine-point circle. Not counting twice the points on each nine-point circle that adjacent faces share, there are a total of 24 points. Amazingly, in an orthocentric tetrahedron, these 24 points are co-spherical!

The center of the 24-point sphere lies on the Eüler segment, and coincides with the centroid of the tetrahedron!

“The real voyage of discovery lies not in finding new lands, but in seeing with new eyes.” Marcel Proust, French novelist/philosopher 1871-1922

References • Altshiller-Court, Nathan. Modern Pure Solid Geometry. The Macmillan Company, 1935. • West, Stephen. Discovering Theorems Using Cabri 3-D. A summary by Ilene Hamilton of a dinner talk given to the Metropolitan Mathematics Club of Chicago, October 3rd, 2008 in Points & Angles, Newsletter of the Metropolitan Mathematics Club of Chicago, Volume XLIII No. 3, November 2008. • Tetrahedron. http://en.wikipedia.org/wiki/Tetrahedron

Does a Tetrahedron Have an Eüler Line?