5. Describe a circle of given radius that shall touch two given lines. How many solutions? 6. Find the locus of the centres of a system of circles touching two given lines. 7. Describe a circle of given radius that shall touch a given circle and a given line, or that shall touch two given circles. PROP. XVII.—Problem. From a given point (P) without a given circle (BCD) to draw a tangent to the circle. Sol.—Let O (fig. 1) be the centre of the given circle. Join OP, cutting the circumference in C. With O as centre, and OP as radius, describe the circle APE. Erect CA at right angles to OP. Join OA, intersecting the circle BCD in B. Join BP; it will be the tangent required. Dem.—Since O is the centre of the two circles, we have OA equal to OP, and OC equal to OB. Hence the two triangles AOC, POB have the sides OA, OC in one respectively equal to the sides OP, OB in the other, and the contained angle common to both. Hence [I. iv.] the angle OCA is equal to OBP; but OCA is a right angle (const.); therefore OBP is a right angle, and [xvi.] PB touches the circle at B. Cor.—If AC (fig. 2) be produced to E, OE joined, cutting the circle BCD in D, and the line DP drawn, DP will be another tangent from P. Exercises. 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. 10. Draw a common tangent to two circles. Hence, show how to draw a line cutting two circles, so that the intercepted chords shall be of given lengths. PROP. XVIII.—Theorem If a line (CD) touch a circle, the line (OC) from the centre to the point of contact is perpendicular to it.