How many triangles!

We can pick 3 of the 100 points to make a triangle, and each of them is distinct, which is equivalent to 100 choose 3.

So the final result is $\displaystyle {100 \choose 3} = \dfrac{100×99×98}{3×2×1}=\boxed{161700}$

We need $3$ points in the plane to make one triangle. Let's suppose we have $6$ points: $A$ , $B$ , $C$ , $D$ , $E$ , $F$

We can rename each point we choose to form the triangle with $Y$ (yes) and all the others with $N$ (no). For example, we can have

(don't care about the numbers down the letters, just look up those ones)

We can form a word putting all the points in a row: $YNNYYN$ . We can just count how many anagrams of the word there are: $\frac{6!}{3! \times 3!}$ $=20$ total anagrams. We have just to apply the same method with $100$ points: the total number of anagrams of a word with $100$ letters, of wich $97$ are $N$ and $3$ are $Y$ is:

$\frac{100!}{97! \times 3!}$ $=$ $\frac{100 \times 99 \times 98}{6}$ $=$ $\boxed{161700}$