  We show that if $A$ and $B$ are finite sets of real numbers, then the number
of triples $(a,b,c)\in A\times B\times (A\cup B)$ with $a+b=2c$ is at most
$(0.15+o(1))(|A|+|B|)^2$ as $|A|+|B|\to\infty$. As a corollary, if $A$ is
antisymmetric (that is, $A\cap(-A)=\est$), then there are at most
$(0.3+o(1))|A|^2$ triples $(a,b,c)$ with $a,b,c\in A$ and $a-b=2c$. In the
general case where $A$ is not necessarily antisymmetric, we show that the
number of triples $(a,b,c)$ with $a,b,c\in A$ and $a-b=2c$ is at most
$(0.5+o(1))|A|^2$. These estimates are sharp.
