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We assume that all the circles in the plane are each colored with one of two colors: red or blue.

My question 1. Does there always exist an equilateral triangle such that its circumcircle and its incircle have the same color?

My question 2. Does there always exist an equilateral triangle such that its incircle and its three excircles have the same color?

My question 3. Does there always exist a regular polygon such that its circumcircle and its incircle have the same color?

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    $\begingroup$ Of course these things are all possible. Do you mean to ask whether these things are necessary? $\endgroup$ – Gerry Myerson Jul 5 '18 at 22:56
  • $\begingroup$ I am sorry If my question is trivial, I only would like to know and to find proof for this. $\endgroup$ – Tran Quang Hung Jul 5 '18 at 23:16
  • $\begingroup$ Pick an equilateral triangle. Color its circumcircle red. Color its incircle red. Color all the other circles any way you like. You have now colored all the circles in the plane, each with one of the two colors, red or blue, and there is an equilateral triangle such that its circumcircle and its incircle have the same color. There is your proof that it is possible to get an equilateral triangle such that its circumcircle and its incircle have the same color. $\endgroup$ – Gerry Myerson Jul 5 '18 at 23:27
  • $\begingroup$ Dear Mr Gerry Myerson. Give an equilateral triangle I can color its circumcircle red but its incircle still possible to be colored blue because all circle have two colors. So my question "Is there possible to get an equilateral triangle such that its circumcircle and its incircle which have the same color?" I think the answer of Prof. Andreas Blass as following is right and good for me. $\endgroup$ – Tran Quang Hung Jul 5 '18 at 23:33
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    $\begingroup$ Of course it is possible to get an equilateral triangle such that its circumcircle and incircle have different colors. But as I have proved it is also possible to get an equilateral triangle such that its circumcircle and incircle have the same color, which is what your question asks for. $\endgroup$ – Gerry Myerson Jul 5 '18 at 23:39
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For Question 1, notice first that you can color the set of positive real numbers with two colors so that $x$ and $2x$ always have different colors: Color $x$ according to the parity of $\lfloor\log_2x\rfloor$. Now assign to each circle the color that you just gave its radius. Since the circumcircle of an equilateral triangle has twice the radius of the incircle, these circles will always have different colors.

Question 2 seems to have a negative answer for a similar reason.

For Question 3, the answer is affirmative. Given any coloring with two colors, consider three concentric circles with radii $1,\sqrt2,2$. The first and second are the incircle and the circumcircle of a square. The second and third are the incircle and the circumcircle of another square. The first and third are the incircle and the circumcircle of an equilateral triangle. Since two of the three circles must have the same color, at least one of the squares or the equilateral triangle has its incircle and the circumcircle colored the same.

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