[Sum of angles in a triangle. Theorem of the exterior angle.] Counting in two ways

In a square, 20 points were marked and connected by non-intersecting segments to each other and to the vertices of the square, such that the square was divided into triangles. How many triangles were formed?
We will consider the marked points and the vertices of the square as vertices, and the segments connecting them and the sides of the square as edges of a planar graph. For each piece into which this graph divides the plane, we will count the number of edges bounding it, and sum all the obtained numbers. Since each edge separates two pieces, we will end up with twice the number of edges. Since all pieces, except the outer one, are triangles, and the outer piece is bounded by four edges, we have \(3(F-1) + 4 = 2E\), which means \(E = \frac{3(F-1)}{2} + 2\). Notice that the number of vertices in our graph is 24, and we will substitute the number of vertices and edges into Euler's formula (problem 30759): 24 \(\left(\frac{1}{2}(F-1) + 2\right) + F = 2\).

From this, \(F = 43\). Thus, the number of triangles into which the square is divided is 42.

## Answer

42 triangles.