Four-point triangle inequality
= Four-point triangle inequality
For four points $x_1,x_2,y_1,y_2$ in a <metric space>,
$$
d(x_1,y_1)-d(x_1,y_2)\leq d(y_1,y_2)\leq d(x_2,y_1)+d(x_2,y_2).
$$
Both steps follow from the <triangle inequality>. The first also holds with an absolute value around the difference, by the <reverse triangle inequality>. This transports a difference of distances measured from one point into a sum measured from another.