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Problem 7 (Triangle Problem)

Any triangle can be divided into similar subtriangles. Show that it is possible to subdivide any triangle into 1999 similar triangles

Solution
A well-known property from geometry states that by drawing the three midlines of a triangle (lines connecting the center points of adjacent sides), we obtain four subtriangles, each similar to the original triangle. See the drawing below. (The similarity is not difficult to verify and comes from the congruence of the corresponding angles.)

If we now repeat this procedure for any of the subtriangles (doesn't matter which one), we get three more subtriangles. If we repeat this procedure on any subtriangle, we obtain three new subtriangles at each step. Hence, we have 1, 4, 7, 10, 13, ... similar subtriangles. So, the equation we ask is: does the equation 1+3k = 1999 have a positive integer solution. Solving this equation, we find k = 666. Hence, after 666 steps, we have 1999 similar triangles of various sizes. Note: The size of these subtriangles depends on which subtriangle you decide to subdivide at each step ... there are all kinds of different sizes. You might play around by drawing different subdivisions yourself.

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Last modified on Tuesday, January 12, 1999