   SEARCH HOME Math Central Quandaries & Queries  Question from Bharathi, a student: given a circle with radius r and a point x,y on its circumference,output two other points x1,y1 and x2,y2 on the circle so that all 3 points form a equilateral triangle. i want a clear explanation please Hi Bharathi,

I presume you have the center of the circle (h, k) as well. Here's how I would do this. The gist is to convert it to a polar co-ordinate system, rotate to find the other two points, then convert back to cartesians.

1. Apply the transformation x x - h and y y - k to slide the centre to the origin.
2. Switch to polar coordinates (r, θ) using the relationship θ = tan-1(y/x).
3. Apply two 120o angular transformations to find the two other points in polar form (r, θi).
4. Convert both new points back to the cartesian coordinates using the relationships xi = r cos(θi) and yi = r sin(θi).
5. Apply the transformation xi xi + h and yi yi + k for both points to move the centre back.

Hope this helps,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.