🚀 go-pugleaf

In article <1993Jan11.084449.15502@Csli.Stanford.EDU> hiraga@Csli.Stanford.EDU (Yuzuru Hiraga) writes:
>In article <1993Jan9.030824.13624@Csli.Stanford.EDU> I wrote
>>A plane is totally colored with two colors.
>[...]
>* Does there exist an isosceles triangle ABC with AB=AC=1cm?

... with A, B, C all the same color, I presume.

The answer is, "Yes."  Suppose not.  Let OPQ be an equilateral
triangle of side 1 cm.  At least two of the vertices must be of the
same color, WLOG suppose O and P are both red.  Now every point R
other than P on the circle of radius 1 cm about O must be blue,
otherwise OPR is the forbidden isoceles triangle.  But then it is easy
to construct a forbidden isoceles triangle (with blue vertices) using
any point S on this circle that is *not* 1 cm away from P, along with
the two distinct points T and U on the circle such that ST=SU=1 cm.

-- David Karr (karr@cs.cornell.edu)