1. An angle inscribed in a semicircle is a right angle. This corollary is mentioned by Aristotle and is attributed
to Thales, being one of the few propositions with which his name is connected. It enables us to describe a circle by letting the arms of a carpenter's square slide along two nails driven in a board, a pencil being held at the vertex.
A more practical use for it is made by machinists to determine whether a casting is a true semicircle. Taking a carpenter's square as here shown, if the vertex touches the curve at every point as the square slides around, it is a true semicircle. By a similar method a circle may be described by sliding a draftsman's triangle so that two sides touch two tacks driven in a board.
Another interesting application of this corollary may be seen by taking an ordinary paper protractor ACB, and fastening a plumb line at B. If the protractor is so held that the plumb line cuts the semicircle at C, then AC is level because it is perpendicular to the vertical line BC. Thus, if a class wishes to determine the horizontal line AC, while sighting up a hill in the direction AB, this is easily determined without a spirit level.
It follows from this corollary, as the pupil has already found, that the mid-point of the hypotenuse of a right triangle is equidistant from the three vertices. This is useful in outdoor measuring, forming the basis of one of the best methods of letting fall a perpendicular from an external point to a line.
Suppose XY to be the edge of a sidewalk, and P a point in the street from which we wish to lay a gas pipe perpendicular to the walk. From P swing a cord or tape, say 60 feet long, until it meets XY at A. Then take M, the mid-point of PA, and swing MP about M, to meet XY at B. Then B is the foot of the perpendicular, since ∠PBA can be inscribed in a semicircle.