The circumradius of a triangle measures 18 cm. What is the radius of its nine-point circle?

The nine-point circle is the image of the circumcircle under the halving enlargement centred at the orthocentre, so its radius is always exactly half the circumradius: 18/2 = 9 cm. The trap 6 cm divides by 3, borrowing the 2 : 1 ratio that belongs to the centroid on a median; 36 cm doubles instead of halving; and 18 cm forgets that the two circles differ at all.