By Alexander Soifer

This moment variation of Alexander Soifer’s How Does One minimize a Triangle? demonstrates how various components of arithmetic should be juxtaposed within the answer of a given challenge. the writer employs geometry, algebra, trigonometry, linear algebra, and jewelry to improve a miniature version of mathematical learn.

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The judges liked the problem. They selected the critical part of it for the juniors (ninth graders) competition: Can every triangle be cut into five triangles similar to each other? Then came the meeting to approve the problems with the Chairman of the Organizing Committee, Andrej Nikolaevich Kolmogorov, one of the greatest mathematicians of the twentieth century. Kolmogorov quietly listened to the presentation of all the problems and their solutions, and then said: 2 How Does One Cut a Triangle?

The middle piece of the partition in Figure 2 How Does One Cut a Triangle? 1 we already cut off the angles of T three triangles similar to T. 8). 2. If a triangle is cut into two triangles T1 and T2 similar to each other, then T1 and T2 are right triangles. 22 2 How Does One Cut a Triangle? 7, then it must have been cut into two triangles T1 and T2 similar to each other. 2, T1 and T2 are right triangles. 1, all five triangles of the partition, as well as the original triangle T, are right triangles.

If the characteristic values λ1 , λ2 , and λ3 of a 3 × 3 matrix A are distinct, then the corresponding characteristic vectors → → → v1 , v2 , and v3 are linearly independent, and therefore form a basis of R3 . Proof. We are given precisely the following three equalities with → → → non-zero vectors v1 , v2 , and v3 : → → → → A v1 = λ 1 v1 (19) A v2 = λ 2 v2 → A v3 = → λ 3 v3 . We need to prove that if → → → → C1 v1 + C2 v2 + C3 v3 = 0 , (20) where C1 , C2 , C3 are real numbers, then necessarily C1 = C2 = C3 = 0.