Finding the Smallest Triangle
Back to Sketchpad at MAA Meetings '96
Find the inscribed triangle of minimum perimeter for a given triangle ABC.
This problem is ideal for demonstrating the power of a dynamic geometry program like the Geometer's Sketchpad. The solution utilizes an "unfolding" technique by taking reflections about the sides of the given triangle ABC.
It is interesting to note that this same technique has been used recently to solve a long standing problem, see G. W. Tokarsky, "Polygonal Rooms Not Illuminable from Every Point," Amer. Math. Monthly, vol.102, 1995, pp. 867-879.
The problem is solved in two steps:
- Find the inscribed triangle of minimum perimeter for a fixed vertex on a side of a given triangle ABC.
- Find the position for the fixed vertex D from Part a which gives an inscribed triangle of minimum perimeter for the given triangle ABC.
First, open the sketch: Min. Inscribed Triangle - Sketch
You will be prompted to choose 3 vertices of a triangle ABC as a starting point for the construction. It is best to start with an acute triangle the first time in order to "see" the technique being presented more clearly. A triangle with a right angle or obtuse angle may give a degenerate inscribed triangle as a solution.
Second, open and run the script: Min. Inscribed Triangle - Script to see the construction of the minimum perimeter triangle for the triangle ABC and a "fixed" vertex D of the arbitrary inscribed triangle (the vertex at the end of the blue bold line).
You will get a construction that looks something like this:
Now, to get the solution for Part a, move the vertices of the inscribed triangle to line up the "unfolded" inscribed triangle (blue bold lines) with the red dashed line segment (compare the given measurement of the perimeter of the inscribed blue triangle with the given measurement of the red dashed line segment). Why does this determine the solution to Part a?
Move the vertices of triangle ABC to see that the solution is an invariant of the construction (and to see what happens when the angles of triagle ABC are not acute).
Third, to determine the solution of Part b, move the "fixed" vertex D of the inscribed triangle from Part 1 above to see that the minimum perimeter triangle occurs at the altitudes of triangle ABC (hint: to see why this is so, consider the isosceles triangle with base the red dashed line DD" and vertex the vertex of triangle ABC which is opposite the side containing the "fixed" vertex D. Then move the "fixed" vertex D to minimize the length of the red dashed line!).
See the sketch: Min. Inscribed Triangle - Sample for a sample triangle and its solution.
This is Problem 75, pp. 85 & 167-170, in I. M. Yaglom, "Geometric Transformations II," MAA New Math. Library #21, 1968.
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