How to find the number of triangles in this figure?

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What is the way to find the number of triangles in the figure below? What is the method or logical way to derive this? I am not looking for formulas but principle that will help me to conquer other shape counting problems i.e. more complex ones?

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3 Answers

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Your figure consists of segments of $5$ straight lines. Since a triangle is determined by the $3$ straight lines forming its sides, the number of triangles is at most $\binom53.$ Since each pair of line segments intersects and all the intersections are distinct, each set of $3$ lines determines a triangle, so the number of triangles is exactly $\binom53=10.$

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First note that every triangle includes at least one point of the star. Because of the symmetry you can just count the triangles that include one point, then multiply to get the rest. Choosing one point, there are three triangles it is part of. One is outside the central pentagon. The other two use an opposite point of the star. The outer triangles just use one outer point, so we have counted them once and there are five total. The spanning triangles include two outer points, so we have counted each one twice. Multiplying $2$ by $\frac 52$ gives five of those as well, for a total of $10$.

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Each vertex of the star can be a vertex of $2$ triangles in one direction (left or right): small and large.

The star has $5$ vertices, hence: $2\cdot 5=10$.

Alternatively: there is a pentagon at the center. There are $5$ small triangles on the sides of the pentagon. There are $5$ large triangles on the vertices of the pentagon. Hence: $5+5=10.$

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