At the cartooning contest, you could draw out your pen and draw a narwhal, and the competition could still end in a draw. The meanings of draw include "to pull gently" and "to sketch" and "an even score".
The original meaning of draw comes from the Old English dragan, "drag." This meaning is still common today, like when you draw the curtains shut or when a horse draws a carriage through Central Park. The most common way to use the verb draw is to mean "produce an image using lines," which came from the idea of drawing a pen or pencil across paper. Occasionally, draw gets confused with drawer, a noun meaning "a compartment that slides out."