TITLE: Prove there are 3 points on the circle having same colour
QUESTION [7 upvotes]: All the points of a circle are randomly coloured red or blue. Prove there
  are 3 points on the circle having same colour, representing an
  isosceles triangle.

REPLY [2 votes]: Not anywhere near as elegant as the five-equidistant-point proof, but I've a nine-point brute-force proof.
Premise: any point on a circle will form an icoseles triangle with points placed equidistantly either side.
Pick any number of points separated by distance X.
If any three sequential points are red (111) or blue (000) then there's an isosceles triangle between those three points, of length 2X. So we need a sequence with no more than two of each color, ever.
But by the same logic, any sequence that repeats with period P, will have an isoceles triangle by the beginning of the third repeat, of length 2PX.
Which means no short sequence can possibly be made which, when repeated, prevents the triangle. 
So if we find the longest binary number made of alternating (1 or 11) and (0 or 00) where no element appears three times separated by the same interval, and we find it's infinitely long, then we've got a winner. This seems unlikely.
We'll start with all 8 possible combinations of "1, (0 or 00), (1 or 11), (0 or 00), 1", and add more to each until they repeat, or it becomes obvious that they will not.
1 10101 - repeat, 1x1x1.

2 101001 -
    10100101 -
      101001010 - repeat, xxxx0x0x0.
      101001011 - repeat, xx1xx1xx1.
    10100100 - repeat, x0xx0xx0.
    10100110 - repeat, x0xx0xx0.

3 1011001 -> repeat, 1xx1xx1

4 101101 - 
    10110101 - repeat, xxx1x1x1
    10110100 - repeat, x0xx0xx0

5 100101 - mirror of 101001.

6 1001001 - repeat, 1xx1xx1.

7 1001101 - mirror of 1011001: repeat, 1xx1xx1.

8 10011001 -
    100110010 - repeat, xx0xx0xx0
    100110011 - repeat, 1xxx1xxx1

There is no binary number longer than 9 digits in which neither 1 or 0 appears twice at a set interval. Which means that picking nine equidistant points around the circle is sufficient to prove the principle.