EMT 668
Write-up III
by Beth Richichi and Brian Seitz

The following is an exploration of triangles using Geometer's SketchPad (GSP). We are attempting to construct various triangles, then construct other geometric constructions on their given sides. This exploration would be extremely difficult without the advantge of having GSP as a tool. The GSP file for this construction can be recalled by clicking here

The original triangle is ABC. An equilateral triangle has been drawn corresponding to each side A, B, and C. In each of these equilateral triangles, the centroid was found and labeled. A' is the centroid of triangle BCJ. It is opposite the A vertex. Each centroid has been connected to its opposite vertex in the original triangle ABC. The point of concurrency inside triangle ABC is X. After drawing the orthocenter (H), circumcenter (c), centroid (G), and incenter (I) of original triangle ABC, it is found that none of these points are the same as X. Point X is its own unique point and does not correspond to any of the other four. Connecting A', B', and C', constructs a fourth equilateral triangle (yellow shaded area). Connecting each centroid to its two adjacent vertexes gives three distinct isosceles triangles. Triangle A'BC (purple area), triangle C'BA (light blue area), and triangle B'AC (green area). Also, a relationship was found when summing the lengths of the three centroids to the opposite vertexes (i.e., AA', BB', CC') and summing the lengths of the three vertexes to point X (i.e., XA + XB + XC). This ratio of the sums is always 2.00. To test any of these ideas hit above and feel free to move the vertexes of triangle ABC as you please. If either angle ABC, angle ACB, or angle CAB are greater than 150 degrees, the point X lies outside of the original triangle ABC This occurs when the centroids lie on a line with two points of original triangle ABC. For example, when A', B, and A are on a line ( with B between A and A') the point X lies on this line as well and the angle is 150 degrees. If original triangle ABC is constrcuted as an equilateral trianlge, then point X X and the four centers of the triangle ALL occur at the same place. This can be best illustrated by clicking above and rotating points A, B, or C. Observe the angle measurements and point X. The GSP file for this construction can be recalled by clicking here.

This second construction is a construction of a triangle ABC with squares constructed on each side of the triangle. The sides of the squares are equal to the corresponding sides of the original triangle. A', B' and C' are the centers of the three squares, respectively. Constructing segments from A' to A, B' to B, and C' to C (red dashed lines) gives us a point of concurrency inside triangle ABC. Again, this point is not the same point as the orthocenter, cetroid, circumcenter, or incenter of the original triangle ABC. Connecting the three points A', B' and C' to form a triangle does not form an equilateral triangle, or one of any significance to this construction. Three distinct isosceles triangles are formed from construction of a center of one of the squaresand the two adjacent points of the original triangle ABC. Triangle ABC', triangle BCA', and triangle ACB' are all isosceles with base equal to a side of the original triangle ABC. Click above to play with and explore this construction. The GSP file for this construction can be recalled by clicking here

Constructing triangle ABC with equilateral triangles on each side (same as first construction) gives a construction as the one above. Labeling A', B', and C' as opposite vertices of the original triangles and drawing segments from A' to A, B' to B, and C' to C gives three dashed lines with a point of concurrency inside original triangle ABC. The three segments are of equal length. Again, the point of concurrency is different from the orthocenter, incenter, centroid, and circumcenter of original triangle ABC. The point of concurrency is inside triangle ABC when angle B'CB creates a positive angle with respect to BC. This is true for all three angles. In this triangle, when angle ABC, angle BAC, or angle ACB are greater than 120 degrees, the point X is outside the original triangle ABC. This occurs becuase an equilateral triangle has three congruent interior angles of 60 degrees, and therefore all supplementary angles will equal 120 degrees. of This can be better understood by clicking above and rotating points A, B, and C and observing the angle values and point X. The GSP file for this construction can be recalled by clicking here

Constructing three isosceles triangles with height equal to the corresponding side of triangle ABC (yellow area) gives us the above representation. Labeling the opposite vertices A', B', and C' and drawing segments from A to A', B to B', and C to C' gives us the following picture. Point X is the point of concurrency for the three segments. Once again, point X is unique from the orthocenter, incenter, circumcenter, and centroid of triangle ABC. Three isosceles triangles have already been constructed (blue, green and orange shaded areas) and no other significant triangles were found. The GSP file for this construction can be recalled by clicking here

This construction is the same as above, but the centroids of the three isosceles triangles have been constructed and designated A', B', and C', respectively. Segments have been constructed from A' to A, B' to B, and C' to C. Point X is the intersection of the three segments. Three unique isosceles triangles are formed from points A', B', and C' to their adjacent vertexes. Triangle A'BC, triangle B'AC, and triangle C'AB are all isosceles with a base equal to a side of the original triangle. The GSP file for this construction can be recalled by clicking here

This construction is a difficult one to view here and is much easier to understand by clicking above, but we will try to explain nonetheless. ABC is the original triangle and three equilateral triangles have been constructed facing the interior of each side. A', B', and C' are the centroids of the three constructed equilateral triangles. Constructing segments from A' to A, B' to B, and C' to C gives us three segments that do not neccessarily intersect at a common point. Selecting point A, B, or C and moving around, sometimes the points intersect. If the three points are made into an equilateral triangle, then all 4 triangles overlap with A', B' and C' being concurrent at the centroid, orthocenter, incenter and circumcenter of triangle ABC.

In conclusion, constructions of various triangles and squares on the sides of an original triangle is an exercise that is highly useful to illustrate properties of triangles. Concepts such as the 4 centers of a triangle, isosceles triangles, equilateral triangles, altitude, and points of concurrency are all developed and discussed. Explorartions with the GSP allow students to visualize the arguments made, that otherwise would be very difficult to do. Without GSP, one would have to construct hundreds of triangles to reach solutions, but with GSP a student can move points as they please and attempt to draw conclusions based on their explorations.

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