MHB Can a Triangle Be Divided Into 205 Congruent Triangles?

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A triangle can indeed be divided into 205 congruent triangles, as demonstrated through geometric principles. The process involves using specific methods of triangulation and symmetry to ensure each resulting triangle maintains equal area and shape. The hint suggests exploring the properties of angles and side lengths to achieve the desired congruence. This division can be achieved through various configurations, including subdividing the triangle into smaller sections that maintain proportional dimensions. Ultimately, the existence of such a division is confirmed through mathematical proof.
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Prove that there exists a triangle which can be cut into 205 congruent
triangles.
 
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Albert said:
Prove that there exists a triangle which can be cut into 205 congruent
triangles.
hint:
Suppose that one side of a triangle has length $n$. Then it can be cut into $n^2$
congruent triangles which are similar to the original one and whose corresponding sides to
the side of length $n$ have lengths 1.
$205=a^2+b^2$ here $a,b\in N$
find all possible $a$ and $b$
 
Thread 'Erroneously finding discrepancy in transpose rule'

Thread 'Erroneously finding discrepancy in transpose rule'

Obviously, there is something elementary I am missing here. To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix in the real plane; taking the transpose we get which then corresponds to a-bi...
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