PROPOSITION XXVI. PROBLEM 391. Through a given point to draw a tangent to a given circle. When the given point is on the circumference. Let A be the given point on the circumference of the circie whose center is 0. Required to draw a tangent to the circle through 4. CASE II When the given point is with out the circumference. Let A be the given point without the circle whose center is 0. Required to draw a tangent to the circle through 4. Draw 04. On 04 as a diameter, describe a circumference, cutting the given circumference at B and C. Draw AB and AC. AB and AC are the required tangents. Draw the radii OB and oc. 1 is a right angle. (?) AB is tangent to the circle. (?) Similarly, 4C is tangent to the circle, Q.E.F CASE II. Second Method 392. With A as center and 40 as a radius, describe the arc DE. With o as a center and the diameter of the given circle as a radius, describe an arc cutting DE at B. Draw OB intersecting the given circle at C. Draw AC. Then AC is the required tangent. [The proof is left for the student.] 393. COROLLARY. The two tangents drawn from a point to a circle are equal; and the line joining the point with the center of the circle bisects the angle between the tangents, and also bisects the chord of contact (BC in the figure to first method) at right angles. 394. SCHOLIUM. When the given point is without the circle, two tangents can be drawn; when it is on the circumference, one, and when it is within the circle, none. 395. DEFINITION. A polygon is circumscribed about a circle when each of its sides is tangent to the circle. In this case the circle is said to be inscribed in the polygon. 396. EXERCISE. If a quadrilateral is circumscribed about a circle, the sum of one pair of opposite sides is equal to the sum of the other pair. Suggestion. Use § 393. 397. EXERCISE. From the point A two tangents AB and AC are drawn to the circle whose center is 0. At any point D on the included arc BC, a third tangent FE is drawn. Prove that the perimeter of the ▲ AEF is constant, and equal to the sum of the tangents AB and AC. E A 399. On a given line to construct a segment that shall contain a given angle. Let AB be the given line and M the given angle. Required to construct on AB a segment that shall contain M. Draw CD through B, making ≤1 = 2 M. Erect BE to CD and bisect AB by the 1 FG. Show that is equally distant from A and B. With o as a center describe a circle passing through A and B. DC is tangent to this circle. (?) 21 AB. (?) Inscribe any angle as ▲ ASB in the segment ARB. LASB ~ AB. (?) ▲ ASB = 21=ZM. (?) The segment ARB is the required segment, since any angle inscribed in it is equal to M. Q.E.F. 400. EXERCISE. tain an angle of 135°. On a given line construct a segment that shall con 401. EXERCISE. What is the locus of the vertices of the vertical angles of the triangles having a common base and equal vertical angles? 402. EXERCISE. Construct a triangle, having given the base, the vertical angle, and the altitude. 403. EXERCISE. Construct a triangle, having given the base, the vertical angle, and the medial line to the base. EXERCISES 1. Two secants, AB and AC, are drawn to the circle, and AB passes through the center. Prove AB> AC. D B 2. One angle of an inscribed triangle is 42°, and one of its sides subtends an arc of 110°. Find the angles of the triangle. 3. Two chords drawn perpendicular to a third chord at its extremities are equal. [Show that BC and AD are diameters, and that A ABC and ADB are equal.] 4. AB and CD are two chords intersecting at E, and CE BE. = 6. What is the locus of the centers of circles tangent to a line at a given point? 7. Pis any point within the circle whose center is O. Prove that PA is the shortest line and PB the longest line from P to the circumference. 8. If a circle is described on the radius of another circle as a diameter, any chord of the greater circle drawn from the point of contact is bisected by the circumference of the smaller circle. A 9. If from a point on a circumference a number of chords are drawn, find the locus of their middle points. (Ex. 8.) 10. From two points on opposite sides of a given line, draw two lines meeting in the given line, and making a given angle with each other. (§ 399.) 11. Work Ex. 10, taking the two points on the same side of the given line. When is the problem impossible? 12. One of the equal sides of an isosceles triangle is the diameter of a circle. Prove that the circumference bisects the base. [Show that BD is 1 to AC.] 13. What is the locus of the centers of circles having a given radius and tangent to a given line? B 14. Describe a circle having a given radius and tangent to two nonparallel lines. How many circles can be drawn? 15. What is the locus of the centers of circles having a given radius and tangent to a given circle? 16. Describe a circle having a given radius and tangent to two given circles. |