10. If two arcs of the same circle or of equal circles are equal, the arcs have equal central angles; and if two minor arcs are unequal, the greater arc has the greater central angle.

If the vertex of an angle is at the center of a circle, and the sides are radii of the circle, the angle is a central angle. If the vertex of the angle is on the circle, and the sides are chords of the circle, the angle is an inscribed angle.

Of the two arcs cut off on a circle by a chord, the smaller is called the minor arc and the larger the major arc. The chord is usually regarded as belonging to the minor arc.

11. If two arcs of the same circle or of equal circles are equal, the arcs have equal chords; and if two minor arcs are unequal, the greater arc has the greater chord.

12. If two chords of the same circle or of equal circles are equal, the chords have equal arcs; and if two chords are unequal, the greater chord has the greater minor arc. 13. If a diameter is perpendicular to a chord, it bisects the chord and its two arcs.

14. If a diameter bisects a chord which is not itself a diameter, it is perpendicular to the chord.

15. The perpendicular bisector of a chord passes through the center of the circle and bisects the arcs of the chord.

16. Equal chords of the same circle or of equal circles are equidistant from the center.

17. Chords that are equidistant from the center of a circle are equal.

18. The less of two chords of the same circle or of equal circles is more remote from the center.

19. If two chords of the same circle or of equal circles are unequally distant from the center, the chord more remote is the shorter.

17. Tangents. 1. A tangent to a circle is an unlimited straight line which touches the circle at only one point. 2. Tangent circles are circles which are tangent to the same line at the same point.

The circles may be tangent externally or internally.

3. If a line is perpendicular to a radius at its end on the circle, the line is tangent to the circle.

4. If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of contact.

5. If a line is perpendicular to a tangent at the point of contact, it passes through the center of the circle.

6. The tangents to a circle from an external point are equal, and make equal angles with the line joining the point to the center.

7. If two parallel lines intersect a circle or are tangent to it, they intercept equal arcs.

18. Angle and Arc Measure. 1. A central angle is measured by its intercepted arc.

2. An inscribed angle is measured by half its intercepted arc.

3. An angle inscribed in a semicircle is a right angle. 4. Angles inscribed in the same segment or in equal segments are equal.

5. An angle formed by a tangent and a chord drawn from the point of contact is measured by half its intercepted arc.

6. An angle formed by two chords intersecting within a circle is measured by half the sum of its intercepted arc and that of its vertical angle.

7. An angle formed by two secants, by a secant and a tangent, or by two tangents drawn to a circle from an external point is measured by half the difference between its intercepted arcs.

19. Loci. 1. To prove that a line or group of lines is a locus we must show that (a) every point on the line or lines satisfies the given condition, and (b) every point which satisfies the given condition lies on the line or lines.

2. The locus of points at a given distance d from a given line x is a pair of lines, I and l', parallel to x and at the distance d from it.

3. The locus of points equidistant from a given point is a circle whose center is the given point.

4. The locus of points equidistant from two points is the perpendicular bisector of the line segment joining them.

5. The locus of points equidistant from two given intersecting lines is a pair of lines which bisect the angles formed by them.

20. Centers of a Triangle. 1. The bisectors of the angles of a triangle meet in a point (the incenter) equidistant from the three sides.

2. The perpendicular bisectors of the sides of a triangle meet in a point (the circumcenter) equidistant from the vertices.

3. The altitudes of a triangle meet in a point (the orthocenter).

4. The medians of a triangle meet in a point (the centroid) which is two thirds of the distance from each vertex to the midpoint of the opposite side.

A median joins the midpoint of a side to the opposite vertex.

22. Numerical Relations in the Triangle. 1. The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

2. The square of either side of a right triangle is equivalent to the difference between the square of the hypotenuse and the square of the other side.

3. The perpendicular from the vertex of the right angle to the hypotenuse of a right triangle is the mean proportional between the segments of the hypotenuse.

b b is the mean proportional between a and c.

b

4. If through two sides of a triangle a line is constructed parallel to the third side, it divides the two sides proportionally.

5. One side of a triangle is to either of its segments cut off by a line parallel to the base as the third side is to its corresponding segment,

6. If a line divides two sides of a triangle proportionally from a vertex, it is parallel to the third side.

7. Three or more parallel lines cut off proportional segments on any two transversals.

8. The bisector of an angle of a triangle divides the opposite side into segments which are proportional to the adjacent sides.

9. If the bisector of an exterior angle of a triangle meets the opposite side produced, it divides that side externally into segments which are proportional to the adjacent sides.

10. The square of the side opposite an acute angle of any triangle is equal to the sum of the squares of the other two sides diminished by twice the product of one of those sides and the projection of the other side upon it.

In the AABC the projection of AC upon AB is the segment AD cut off on AB by a perpendicular from C to AB.

11. The square of the side opposite the obtuse angle of any obtuse triangle is equal to the sum of the squares of the other two sides increased by twice the product of one of those sides and the projection of the other side upon it.

12. The sum of the squares of two sides of a triangle is equal to twice the square of half the third side, increased by twice the square of the median upon it.

13. The difference between the squares of two sides of a triangle is equal to twice the product of the third side and the projection of the median upon it.

23. Numerical Relations in the Circle. 1. Two circumferences have the same ratio as the radii.

2. In any circle, C=2πrd,