Ilya's logical tests

This problem was offered at the Moscow City Mathematical Olympiad for high school seniors in 1978.

A policeman is patrolling the streets assigned to him according to the map below, and initially stands at the central intersection. Somewhere on these streets there is also a thief. The policeman cannot see the thief unless located on the same straight line as the thief. It is known that the thief's speed is less than half of the policeman's speed. You are the policeman. Catch the thief. To move, click where you want to go.