Transcript Slide 1

Does a Tetrahedron
Have an Eüler Line?
PCMI Summer 2011
Park City, Utah
Thursday, July 6th, 2011
Troy Jones
Westlake High School, Saratoga Springs, Utah
[email protected]
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Important Vocabulary
• Concur
lines/planes
• Coincide points/lines/planes
• Collinear
points
• Cyclic
points
• Coplanar
points
• Cospherical points
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“Math Curse”, Jon Scieszka, Viking Press, 1995
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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
(a) to construct a plane given three of its points not
lying in a straight line;
(b) to construct the line of intersection of two planes;
(c) to construct the point of intersection of a line and
a plane;
(d) 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.
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What is the locus of balancing points for these thin
strips of paper in the approximate shape of a triangle?
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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.
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Slicing the triangle into strips parallel to any side,
and tracing the midpoints, will create three different
balancing lines, or medians.
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The medians will concur at a point called the centroid,
which will be the balancing point, or center of mass,
for the triangle.
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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?
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We could slice a tetrahedron into thin slices
parallel to a face. Each slice would be a triangle
that balances at its centroid.
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The locus of these centroids would form a
segment connecting a vertex of the tetrahedron
with the centroid of the face opposite the
vertex.
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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.
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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.
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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.
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The locus of the centroid of the parallelogram is the
midsegment of the tetrahedron, connecting the
midpoints of opposite edges of the tetrahedron.
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The three midsegments of a tetrahedron also
concur at the centroid of the tetrahedron. The
centroid is the midpoint of each midsegment.
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The four medians and the three midsegments all
concur at the centroid of the tetrahedron.
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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.
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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.
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The locus of points in a plane equidistant from the
endpoints of a segment form a line called the
perpendicular bisector of the segment.
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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.
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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.
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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.
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Here is just the line of intersection of the
perpendicular bisectors of the edges of face PQR.
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The perpendicular bisectors of the six edges of a
tetrahedron concur at a point called the
circumcenter of the tetrahedron.
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The circumcenter of the tetrahedron is equidistant
from all four vertices, and is the center of the
circumscribed sphere of the tetrahedron.
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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.
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In general, the altitudes of a tetrahedron do not
concur. (Not every tetrahedron has an orthocenter).
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If the altitudes do concur, the point of
concurrency is called the orthocenter of the
tetrahedron, and the tetrahedron is called
orthocentric.
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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.
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Question:
Does a tetrahedron have an Eüler line?
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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.
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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.
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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.
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In an orthocentric tetrahedron, the Monge point and
the orthocenter coincide.
Gaspard Monge
1746-1818
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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.
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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.
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The tetrahedron formed by connecting the
centroids of the faces of a tetrahedron is called the
medial tetrahedron. The circumcenter of the
medial tetrahedron circumsphere is on the Eüler
segment, and it divides the Eüler segment in a 1:2
ratio.
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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.
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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.
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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!
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The center of the 24-point sphere lies on the Eüler
segment, and coincides with the centroid of the
tetrahedron!
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“The real voyage of discovery
lies not in finding new lands,
but in seeing with new eyes.”
Marcel Proust, French novelist/philosopher 1871-1922
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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
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