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 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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