It should be noticed that Euclid speaks of "equal circles," while we speak of "the same circle or equal circles," confining our proofs to the latter, on the supposition that this sufficiently covers the former. THEOREM. _A line through the center of a circle perpendicular to a chord bisects the chord and the arcs subtended by it._ This is an improvement on Euclid, III, 3: "If in a circle a straight line through the center bisects a straight line not through the center, it also cuts it at right angles; and if it cuts it at right angles, it also bisects it." It is a very important proposition, theoretically and practically, for it enables us to find the center of a circle if we know any part of its arc. A civil engineer, for example, who wishes to find the center of the circle of which some curve (like that on a running track, on a railroad, or in a park) is an arc, takes two chords, say of one hundred feet each, and erects perpendicular bisectors. It is well to ask a class why, in practice, it is better to take these chords some distance apart. Engineers often check their work by taking three chords, the perpendicular bisectors of the three passing through a single point. Illustrations of this kind of work are given later in this chapter. THEOREM. _In the same circle or in equal circles equal chords are equidistant from the center, and chords equidistant from the center are equal._ This proposition is practically used by engineers in locating points on an arc of a circle that is too large to be described by a tape, or that cannot easily be reached from the center on account of obstructions. [Illustration] If part of the curve _APB_ is known, take _P_ as the mid-point. Then stretch the tape from _A_ to _B_ and draw _PM_ perpendicular to it. Then swing the length _AM_ about _P_, and _PM_ about _B_, until they meet at _L_, and stretch the length _AB_ along _PL_ to _Q_. This fixes the point _Q_. In the same way fix the point _C_. Points on the curve can thus be fixed as near together as we wish. The chords _AB_, _PQ_, _BC_, and so on, are equal and are equally distant from the center. THEOREM. _A line perpendicular to a radius at its extremity is tangent to the circle._ The enunciation of this proposition by Euclid is very interesting. It is as follows: The straight line drawn at right angles to the diameter of a circle at its extremity will fall outside the circle, and into the space between the straight line and the circumference another straight line cannot be interposed; further, the angle of the semicircle is greater and the remaining angle less than any acute rectilineal angle.