Connect Dots To Form Triangles

First by ignoring the question. Lets first find the total number of triangles formed.

$\large \displaystyle \text{Total Triangles Formed } = C \binom{15}{3} = \frac{15!}{12! \times 3!}\\ \large \displaystyle = \frac{15 \times 14 \times 13}{1 \times 2 \times 3} = 455.$

So there are $455$ triangles we can draw using 15 dots on a circle.

And now if we come to equilateral triangle. We know that all the dots are equally spaced. Suppose the points are numbers from 1 to 15. From point 1 to point 6 is one-third of the circle — again, from point 6 to point 11, and from point 11 back to point 1. That’s one equilateral triangle.

So, We could make an equilateral triangle using points $(1,6,11); (2,7,12); (3,8,13); (4,9,14); (5,10,15)$

Therefore, There are only five equilateral triangles formed.

$\large \displaystyle \text{Number of Triangles formed } = 455 - 5 = \color{#D61F06}{\boxed{450}}.$