Math CentralQuandaries & 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.

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