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Let $A_1A_2A_3A_4$ be a square, and let $A_5,A_6,A_7,\ldots,A_{34}$ be distinct points inside the square. Non-intersecting segments $\overline{A_iA_j}$ are drawn for various pairs $(i,j)$ with $1\le i,j\le 34$, such that the square is dissected into triangles. Assume each $A_i$ is an endpoint of at least one of the drawn segments. How many triangles are formed?


how can I approach this problem?

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    $\begingroup$ I'd try smaller numbers of points, first, to see if there is a pattern. $\endgroup$ – Thomas Andrews Jun 25 '15 at 21:46
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Hint:

  • From Euler characteristic we know that $V-E+F=2$.
  • Because the square is triangulated, there is a relation between number of edges and number of faces:

    $$3(F-1)=2E-4$$ (the outer face is a square and we count its edges only once).

I hope this helps $\ddot\smile$

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