The triangle above contains several straight lines that create smaller triangles inside. How many triangles are there altogether? Try to count them before you look at the solution!
We start with the baseline AB shown in blue.
We look at all the triangles with it as base.
We count as follows.
We look at each red line, and we see that on each red line, there are four vertices which are vertices of a triangle with base AB. For example, on line 1B, we have vertices 1, 2, 3, and 4, giving triangles AB1, AB2, AB3, and AB4. So therefore the total number of triangles we have counted so far is 16. Notice that for each of the triangles we have counted so far, the base is blue and one side is green, and the other side is red. Note that one of the 16 is equilateral, and three are isosceles.
Now let’s delete base AB, since we have no more triangles with it as a base.
Notice now that every other triangle contains either A as a vertex, or B as a vertex, but not both A and B. So let’s count all triangles that contain vertex A.
Each of them has two green sides and one red side. So, how many such triangles have their red side on one line?
On each red line, besides vertex B, we have four other vertices. So how many triangles with 2 green sides have a red side on this line?
We can choose two things out of four things in six different ways, so we have six triangles with two green sides that have red sides on one red line. For example, on the bottom red line, we have A12, A13, A14, A23, A24, and A34.
We have 4 red lines, so therefore the total number of triangles with two green sides and one red side is 4 times 6 = 24.
Finally, we count all triangles that have two red sides and one green side. These triangles contain vertex B and they do not contain vertex A. But each such triangle corresponds to exactly one triangle with two green sides and one side, which we have already counted. For example, B7 11 corresponds to A 76, and B6 10 corresponds to A 10 11, and so on. Therefore we have 24 triangles with two red sides and one green side.
If you don’t understand this part of the derivation, you may count the triangles with two red and one green side in the same way as you counted the triangles with two green and one red side.
So there are a total of 16 + 24 + 24 = 64 triangles!
