26. In the coordinate plane, construct a right triangle such that its two legs are parallel to the $x$ and $y$ axes, and such that the midline of one of its legs lies on one of the lines $y=3x+1$ and $y=mx+2$. How many constants $m$ allow for the existence of such a triangle?
(A) 0 ; (B) 1 ; (C) 2 ; (D) 3 ; (E) More than 3.
26
C
26. It is easy to prove using analytic geometry that: "In a right-angled triangle with its two legs parallel to the coordinate axes, the slope of the median on one leg is necessarily 4 times the slope of the median on the other leg." As shown in the figure, a triangle of this kind is drawn, where

$a, b, c, d$ are arbitrary, with $c \cdots$, $d \neq 0$. Note that the slopes of the two medians are
$$
\frac{c}{2 d} \text { and } \frac{2 c}{d}=4\left(\frac{c}{2 d}\right) \text {. }
$$

From this conclusion, in our problem, $m$ can be determined to be 12 or $\frac{3}{4}$. In fact, it can be verified that both values are possible.