A Map of All Triangles*and a search for an ideal acute scalene triangle
A Map of All Triangles*For her geometry students, my wife was cutting out paper triangles and labeling the results of paper folding to find medians, altitudes, perpendicular bisectors, and angle bisectors. When she placed an acute scalene triangle next to me on her desk, I remembered times as a student that I was asked to sketch an acute scalene triangle to investigate some triangle property. And I remember that after establishing a base for the triangle that choosing the third vertex seemed to be a tricky task. It was easy to accidently make the triangle right or isosceles. So I thought why not try to figure out where the traps are beforehand and maybe that will make it possible to construct a very acute and scalene triangle
Starting with a fixed side, I found all the points that, as third vertices, make a right or isosceles triangle. Then, if we avoid these properties we'll have a non-right, scalene triangle. To make an acute triangle, we can use the fact that the plane is divided into acute and obtuse regions by the sets of points which yield a right triangle.
Another observation: given any triangle, a Triangle Figure may be constructed about either of the three sides. The third vertex in each case will land in the same type of region or on the same type of curve. If one vertex lies in an acute region of a Triangle Figure, then so will the others. Each region of the same type, however, corresponds to a different order of the lengths of the sides of triangle ABC. A picture will make all of this much clearer.
My wife, Donna Simms, teaches Geometry for Elementary Education Majors and other math courses here at Clemson.
The next section offers a more rigorous development of the triangle figure.
This
explanation is suitable for any student who has had some geometry lessons in
elementary, middle, or high-school. At four pages, a pamphlet can be made by
placing pages 1 & 4 and 3 & 2 on a photocopier and using 64% reduction and
double-sided settings.
The pictures in this pamphlet were drawn with Geometer's Sketchpad.
A supplement to this introduction is an interactive triangle property explorer. Basically, it's a triangle, a moveable third vertex, and an accompanying triangle figure. You can drag the third vertex around and see what kind of triangle you get relative to the third vertex's location in the triangle figure.
Approaching triangle construction from another point of view, this Java applet allows one to experiment with a triangle determined by one side and the point of intersection of its altitudes. Altitude Manipulator
Charles Dodgeson, a.k.a. Lewis Caroll, posed this question: If three points are chosen at random in a plane, what's the probability that the triangle formed will be obtuse? His solution uses a figure that is a subset of the figure presented here.
Eric Weisstein's CRC Concise Encyclopedia of Mathematics gives the details on this question under the entry for Obtuse Triangle. -- Currently not available on the web.
Let's use the term pencil of triangles for a set of all the triangles that share a specific line segment as a side, which can be called the base of the pencil. Any triangle in a pencil of triangles can be referred to by its third vertex, the one not incident with the base of the pencil.
After designating a triangle, congruent copies may be found elsewhere within the same pencil of triangles. One way is to reflect a triangle in the perpendicular bisector of the base. Another is to reflect a triangle in the line containing the base. These two reflections may be used to generate a total of four congruent triangles within the same pencil, as long as the third vertex of the initial triangle is not on either of the lines of reflection.
If we expand the search to include similar copies of a triangle, then scaling becomes a possibility. We can make another side of a triangle match the base of the pencil with the transformations of rotation, scaling and translation. In most cases, this will result in a different third vertex. Another way to think of this is to permute the angles in a triangle relative to the base. The resulting triangle may be of a different size, but it will have the same three angles, and hence be similar to the original.
Here's a JavaSketchpad-produced applet that shows the third vertices of 5 triangles similar to triangle ABC in the same triangle pencil. C is moveable.
More to be added.
[Preview: transforming third vertices with circle inversion in the circles centered at A and B in the figure is another way to produce the third vertices of the similar triangles mentioned above.]
Overlay a 2-dimensional coordinate system onto the triangle figure with A:(0,0) and B:(1,0) as the endpoints of the base of the pencil of triangles. Let's call this pencil the unit pencil of triangles.
Treating this as the complex plane, we can use functions of complex numbers to find sets of similar triangles within the unit pencil of triangles.
More to be added.
[Preview: the functions f(z)=1-1/z, g(z)=1-Conjugate(z), and h(z)=Conjugate(z) map third vertices to third vertices of similar triangles. f and g applied to any third vertex, except an isosceles triangle's, will produce five additional points. These points are third vertices of triangles similar to the original. The set of six points lie on a circle that is orthogonal to circles centered at A and B in the triangle figure.]
Geoge Pölya, in his book How to Solve It, mentions a 45-60-75 triangle as the most general triangle for use as an example when considering a universal property of triangles. In the triangle figure, it seems like the centers of circles inscribed in the acute subregions would make for very actue and scalene triangles.
Three different sized circles result. The radii of the three circles are 1/4, 1/6, and 1/8. Double these to get the diameters 1/2, 1/3, 1/4 -- which happens to be part of the harmonic sequence. No explanation or use for this has yet been found.
The circle centers used as third vertices of triangles don't yield a set of similar triangles. However, these triangles are very acute and scalene.
[More to be added.]
In Euclidean geometry, circle inversions can be used to make similar triangles that share a side. In the Poincaré Model of Hyperbolic Geometry, circle inersions are used to determine congruence.
Looking for an "ideal" acute scalene triangle led me to the Poincaré model of hyperbolic geometry. This is because all of the parts of lines and parts of circles above line AB could be considered to be h-lines in that model.
This may be a strange area to some, so here's an Introduction to Hyperbolic Geometry (a work in progress).
At first I tried using type-2 h-lines to bisect the angles at two corners of each of the acute subregions. This I thought might be analogous to finding the incenter of a euclidean triangle with angle bisectors. The points found in each subregion gave a set of mutually similar triangles!
After further reading about hyperbolic geometry models, it was found that these points are the hyperbolic centers of the inscribed circles. Compared to the Euclidean centers of the inscribed circles, these points were shifted vertically by a small amount towards h-bar.
One of the points turned out to have an easy shortcut construction, given the base. Its coordinates are (1/4, Sqrt(23)/4) with base vertices at (0,0) and (1,0). This triangle has side lengths of 1, Sqrt(6)/2, and Sqrt(2) and an altitude of Sqrt(23)/4 which divides the base of length 1 into 1/4 and 3/4.
Some properties of this triangle:
* All triangles in the Euclidean plane can be made to share a side by using the transformations of scaling, rotation, and translation. The resulting triangles will be similar to the originals and, hence, be of the same type. Types being the angle properties of acute, right, or obtuse and the side length properties of equilateral, isosceles, or scalene.
Disclaimer: This page is in the process of being brought up to speed with what I've done so far on this project. It does not yet cover everything that I've discovered. This disclaimer may be here for quite some time.
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