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Triangles are used extensively because they are fundamentally rigid, because three line segments can define one and only one triangle.

Compare a trianlge to, say, a square, which could flex at its vertices to form a rhombus. If you take a square, however, and insert one diagonal, you basically have two triangles, which make the square rigid and not prone to collapse.

Arches are also fundamental in architecture because of the way they distribute weight to the pillars that support them. Arches also convert horizontal and lateral forces to vertical ones.

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