20. In a right-angled triangle, having given the sum of the base 21. Given the perimeter of a right-angled triangle whose sides 22. Given the difference of the angles at the base, the ratio of 23. Given the difference of the angles at the base, the ratio of 24. Given the base of a right-angled triangle; to construct it, 25. Given one angle of a triangle, and the sums of each of the 26. Given the vertical angle, and the ratio of the sides contain- 27. Given the vertical angle, and the radii of the inscribed and 28. Given the vertical angle, the radius of the inscribed circle, 29. Given the base, ore of the angles at the base, and the point 30. Given the vertical angle, the base, and the difference between 31. Given that segment of the line bisecting the vertical angle 8. Given one angle, a side opposite to it, and the sum of the other two sides; to construct the triangle. 9. Given the vertical angle, the line bisecting the base, and the angle which the bisecting line makes with the base; to construct the 10. Given the vertical angle, the perpendicular drawn from it to the base, and the ratio of the segments of the base made by it; to 11. Given the vertical angle, the base, and a line drawn from either of the angles at the base to cut the opposite side in a given ratio; to construct the triangle. 12. Given the perpendicular, the line bisecting the vertical angle, and the line bisecting the base; to construct the triangle. 13. Given the line bisecting the vertical angle, the line bisecting the base, and the difference of the angles at the base; to construct 14. Given the vertical angle, and the line drawn to the base bisecting the angle, and the difference between the base and the sum of the sides; to construct the triangle. 15. Given the line bisecting the vertical angle, the perpendicular drawn to it from one of the angles at the base, and the other angle at the base; to construct the triangle. 16. Given the line bisecting the vertical angle, and the perpen- diculars drawn to that line from the extremities of the base; to 17. Given the vertical angle, the difference of the two sides con- taining it, and the difference of the segments of the base made by a perpendicular from the vertex; to construct the triangle. 18. Given the base, and vertical angle; to construct the triangle, when the square of one side is equal to the square of the base, and three times the square of the other side. 19. Given the base and perpendicular; to construct the triangle, when the rectangle contained by the sides is equal to twice the rectangle contained by the segments of the base made by the line g 20. In a right-angled triangle, having given the sum of the base and hypothenuse, and the sum of the base and perpendicular; to construct the triangle. 21. Given the perimeter of a right-angled triangle whose sides are in geometrical progression; to construct the triangle. 22. Given the difference of the angles at the base, the ratio of the segments of the base made by the perpendicular, and the sum of the sides; to construct the triangle. 23. Given the difference of the angles at the base, the ratio of the sides, and the length of a third proportional to the difference of the segments of the base made by a perpendicular from the vertex and the shorter side; to construct the triangle. 24. Given the base of a right-angled triangle; to construct it, when parts, equal to given lines, being cut off from the hypothenuse and perpendicular, the remainders have a given ratio. 25. Given one angle of a triangle, and the sums of each of the sides containing it and the third side; to construct the triangle. 26. Given the vertical angle, and the ratio of the sides containing it, as also the diameter of the circumscribing circle; to construct the triangle. 27. Given the vertical angle, and the radii of the inscribed and circumscribing circles; to construct the triangle. 28. Given the vertical angle, the radius of the inscribed circle, and the rectangle contained by the straight lines drawn from the centre of that circle to the angles at the base; to construct the triangle. 29. Given the base, ore of the angles at the base, and the point in which the diameter of the circumscribing circle drawn from the vertex meets the base; to construct the triangle. 30. Given the vertical angle, the base, and the difference between two lines drawn from the centre of the inscribed circle to the angles at the base; to construct the triangle. 31. Given that segment of the line bisecting the vertical angle which is intercepted by perpendiculars let fall upon it from the angles at the base; the ratio of the sides; and the ratio of the radius of the inscribed circle to the segment of the base which is intercepted between the line bisecting the vertical angle and the point of contact of the inscribed circle; to construct the triangle. 32. Given the line bisecting the vertical angle, and the differences between each side and the adjacent segment of the base made by the bisecting line; to construct the triangle. 33. Given one of the angles at the base, the side opposite to it, and the rectangle contained by the base and that segment of it made by the perpendicular which is adjacent to the given angle; to con struct the triangle. 34. Given the vertical angle, and the lengths of two lines drawn from the extremities of the base to the points of bisection of the sides; to construct the triangle. 35. Given the lengths of three lines drawn from the angles to the points of bisection of the opposite sides; to construct the triangle. 36. Given the segments of the base made by the perpendicular, and one of the angles at the base triple the other; to construct the triangle. 37. The area and hypothenuse of a right-angled triangle being given; to construct the triangle. 38. Given one angle, and a line drawn from one of the others bisecting the side opposite to it; to construct the triangle, when the area is also given. 39. In two similar right-angled triangles, the sum of the base of one and perpendicular of the other is given; to determine the triangles such that their hypothenuses may contain the right angle of another triangle similar to them, and the sum of the three areas may be equal to a given area. 40. Given the vertical angle, the area, and the distance between the centres of the inscribed circle and the circle which touches the base and the two sides produced; to construct the triangle. 41. Given the area, the line from the vertex dividing the base into segments which have a given ratio, and either of the angles at the base; to construct the triangle. 42. Given the difference between the segments of the base made by the perpendicular, the sum of the squares of the sides, and the area; to construct the triangle. 43. Given the base, one of the angles at the base, and the difference between the side opposite to it and the perpendicular; to construct the triangle. 44. Given the vertical angle, the difference of the base and one side, and the sum of the perpendicular drawn from the angle at the base contiguous to that side upon the opposite side and the segment cut off by it from that opposite side contiguous to the other angle at the base; to construct the triangle. 45. Given the base, the difference of the sides, and the segment intercepted between the vertex and a perpendicular from one of the angles at the base upon the opposite side; to construct the triangle. 46. Given the vertical angle, the side of the inscribed square, and the rectangle contained by one side and its segment adjacent to the base made by the angular point of the inscribed square; to construct the triangle. |