9. Two circles intersect at B and C. BA and BD are drawn tangent to the circles. Prove that BC is a mean proportional between AC and CD. [Prove▲ ABC and BCD similar.] 10. Find the lengths of the longest and A the shortest chords that can be drawn through a point 10 in. from the center of a circle having a radius 26 in. 11. Tangents are drawn to a circle at the extremities of the diameters AB. Secants are drawn from A and B, meeting the tangents at D and E and intersecting at C on the circumference. Prove the diameter a mean proportional between the tangents AD and BE. [A ABD and ABC are similar. (?)] B D E 12. If two circles are tangent internally, chords of the greater drawn from the point of tangency are divided proportionally by the circumference of the less. 13. If two circles are tangent externally, secants drawn through their point of contact and terminating in the circumferences are divided proportionally at the point of contact. 14. Given the two segments of the base of a triangle made by the bisector of the vertical triangle, and the sum of the other two sides, to construct the triangle. [§ 502.] 15. Determine a point P in the circumference, from which chords drawn to two given points A and A B B 18. The point of intersection of the medians, the point of intersection of the perpendiculars at the middle points of the sides, and the point of intersection of the altitudes of a triangle are in the same straight line. [See Ex. 17.] 19. The triangles ABC B and ADC have the same base and lie between the same parallels. EF is drawn parallel to AC. Prove EG: = HF. 20. Two tangents are drawn at the extremities of the diameter AB. At any other point C on the circumference a third tangent DE is drawn. Prove that OD is a mean proportional between AD and DE, and that OE is a mean proportional between BE and DE. [Prove DOE a R.A., and use § 518.] 21. The prolongation of the common chord of two intersecting circles bisects their common tangent. [§ 539.] 22. To draw a line AC intersecting two given circles so that the chords AD and BC shall be of given lengths. [See Ex. 24, p. 125.] O B 23. xy is any line drawn through the vertex A of the parallelogram ABCD and lying xwithout the parallelogram. Prove that the perpendicular to xy from the opposite angle C is equal to the sum of the perpendiculars from B and D to xy. [§ 453.] 24. The sum of the perpendiculars from the vertices of one pair of opposite angles to a line lying without a parallelogram is equal to the sum of the perpendiculars from the vertices of the other pair of opposite angles. E 25. Two circles are tangent externally at C. DE and CF are common tangents. Prove that DCE = 1 R.A., and also that AFB = 1 R.A. 26. Prove that A DFC and CBE (see figure of Ex. 25) are similar, as are also ADAC and FCE. E 27. Describe a circle passing through two given points and tangent to a given line. [The line joining the two given points A and B may be parallel to the given line CD (see Fig. 1), or its prolongation may meet the given line (see Fig. 2). In the second case DE2 = DA × DB. (?) DE may be laid off on either side of D, ... two © can be described fulfilling the conditions of the problem.] 33. In the triangle ABC, let two lines drawn from the extremities of the base AC and intersecting at any point D on the median through B, meet the opposite sides in E and F. Show that EF is parallel to AC. 34. ABC is an acute-angled triangle. DEF (called the pedal triangle) is formed by joining the feet of the altitudes of triangle ABC. Prove that the altitudes of triangle ABC bisect the angles of the pedal triangle DEF. [A O can be described passing through F, O, D, and B. (?) 1= 22. (?)] 35. Prove the triangles Afe, bfd, and DCE similar to triangle ABC and A to each other. [See figure of Ex. 34.] [To prove & FBD and ABC similar. Show that A= 22.] 36. Prove that the sides of the triangle ABC [see Ex. 34] bisect the exterior angles of the pedal triangle DEF. [Show 37. The three circles that pass through two vertices of a triangle and the point of intersection of the altitudes are equal to each other. that each is equal to the circle circumscribed about the triangle.] 38. Describe a circle passing through two given points and tangent to a given circle. [A and B are the given points and C the given circle. DEAB is any O passing through A and B and cutting the given O C. The common chord ED meets AB at G. GF is tangent to O C. AFB is the required O.] G B 39. If one leg of a right-angled triangle is double the other, a perpendicular from the right angle to the hypotenuse divides it into segments having the ratio of 1 to 4. 40. The triangle ABC is inscribed in a circle, and the bisector of angle B intersects AC at D and the circumference at E. AB Prove BD BC E |