1. The two tangents PB, PD (fig. 2) are equal to one another, because the square of each is equal to the square of OP minus the square of the radius.
2. If two circles be concentric, all tangents to the inner from points on the outer are equal.
3. If a quadrilateral be circumscribed to a circle, the sum of one pair of opposite sides is equal to the sum of the other pair.
4. If a parallelogram be circumscribed to a circle it must be a lozenge, and its diagonals intersect in the centre.
5. If BD be joined, intersecting OP in F, OP is perpendicular to BD.
6. The locus of the intersection of two equal tangents to two circles is a right line (called the radical axis of the two circles).
7. Find a point such that tangents from it to three given circles shall be equal. (This point is called the radical centre of the three circles.)
8. The rectangle OF.OP is equal to the square of the radius.
Def. Two points, such as F and P, the rectangle of whose distances OF, OP from the centre is equal to the square of the radius, are called inverse points with respect to the circle.
9. The intercept made on a variable tangent by two fixed tangents subtends a constant angle at the centre.