Acute Dissection

 

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The minimal number of acute triangles is 7. The pattern for the seven triangles is shown in the illustration. An elegant proof for the seven acute triangles has been provided by Wallace Manheimer in American Mathematical Monthly, November 1960. The logic behind the proof is as follows.

The obtuse angle must be divided by a line. This line cannot go all the way to the other side, for then it would form another obtuse triangle (or two triangles with right angles), which in turn would have to be dissected, consequently the pattern for the large triangle would not be minimal. The line dividing the obtuse angle must, therefore, terminate at a point inside the triangle. At this vertex, at least five lines must meet, otherwise the angles at this vertex would not all be acute. This creates the inner pentagon of five triangles, making a total of seven triangles as shown in the illustration.

Last Updated: May 16, 2013
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