### Dr. Wilson

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