### Thales of Miletus.

One time, an enemy warship dropped anchor off the coast of
Miletus. Since it would have proved impracticable to take a chain out
to the ship to measure how far off the shore it was, Thales came up
with the following ingenious plan to determine how far off the coast
the ship was anchored. He first laid out a line down the beach from A
to B.

He measured the angle which the line from A to the ship at S made
with the line from A to B, and measured off the same angle going
inland. He did the same thing at point B. He found the point C where
these two lines met. At the point C, he sighted on the ship and found
the point D where the line from C to the ship met the line from A to
B. Since C was inland, he could measure the distance from C back to
the line at D. Prove that D is the closest point from the ship to the
line AB, and that the distance from D to the ship is the same as the
distance from C to D.

Given:

- angle
*BAC* = angle *BAS*
- angle
*ABC* = angle *ABS*

To prove:

*CD* = *DS *
*AB* is perpendicular to *CS *