Prop. 7. LAQF LAQB; Similarly AQBBQC, &c.; ..s at cent. Q. subtended by each side are equal to each other. Similarly, s at cent. R subtended by each .... Z AQF Also side each other. = = /GRM being like parts of 4 Prop. 70. rt. s. the As in each polygon are isosceles, ../sQAF, QFA=/s RGM, RMG ea. to ea. Prop. 5. .. As AQF, GRM are equiangular. .. AQ : GR :: AB + BC + CD, &c. : GH + HI+IK, &c. i.e. AQ GR :: ce of polygon ABC, &c.: ce of polygon GHI, &c. Therefore, the ces &c. COR.-If the sides be indefinitely diminished, and their no. increased, the ces of the polygons will coincide with the ces of the circumscribing Os. the circumferences of circles are to other as each their radii, or as their doubles, viz., their diameters. Prop. 73. Prop. 68. Prop. 70. PROP. LXXXVI. THEOR. The areas of circles are to each other as the squares of their radii, or as the squares of their diameters. (Figs. last prop.) H Prop. 83. Prop. 70. Prop. 70. For AQF: GRM :: AQ2: GR2; and equimultiples of As AQF: GRM have the same ratio; .. Area of polygon ABC, &c. : area of polygon GHI, &c. :: AQ2 : GR2. If the sides be indefinitely diminished, and their number increased, the areas of the two polygons will be equal to the areas of their circumscribing Os; .. Area of ABC, &c.: area of O GHI &c. :: AQ2: GR2, or :: AD2: GK2. Therefore the areas of circles, &c. EXERCISES. 1. In a given straight line, to find a point equally distant from two given points. 2. Given one side of a right-angled triangle, and the difference between the hypothenuse and the other side; show how the triangle can be determined. 3. The straight line which bisects the angle opposite the base of an isosceles triangle, bisects the base also, and is at right angles to it. 4. In a given straight line to find a point from which, if lines be drawn to two given points on the same side of the line, these lines may make equal angles with it. 5. If the base of an isosceles triangle be bisected by a line drawn from the opposite angle, this angle will be also bisected, and the line will be perpendicular to the base. 6. The sum of the perpendiculars drawn from any point in one side of an equilateral triangle upon the other sides, is equal to the perpendicular from either of the angles, to its opposite side. 7. A perpendicular is the least distance of a given point from a given line. 8. The difference between any two sides of a triangle is less than the third side. 9. The difference between the sum of any two sides of a triangle and the third side, is less than twice the line drawn from any point in that side to the angle opposite. 10. If two straight lines bisect two sides of a triangle perpendicularly; the perpendicular from the point of intersection of these lines upon the third side will bisect that side. 11. Through a given point to draw a straight line which shall make equal angles with two given straight lines. 12. A perpendicular to the base of an isosceles triangle to the opposite angle, will bisect that angle and the base. 13. If the base of a triangle be bisected by a straight line drawn from the opposite angle perpendicular to the base, that angle is bisected, and the triangle is isosceles. 14. If the angle of a triangle be bisected by a straight line perpendicular to the opposite side, that side is bisected, and the triangle is isosceles. 15. If an angle of a triangle and its opposite side be bisected by a straight line; the line is perpendicular to the side, and the triangle is isosceles. 16. Trisect a given straight line. 17. Trisect a right angle. 18. The greatest diameter of a parallelogram is that which is opposite to the greatest angle. 19. If the opposite sides of a quadrilateral figure are equal, it is a parallelogram. 20. If the opposite angles of a quadrilateral figure are equal, it is a parallelogram. 21. The difference between the angles at the base of a triangle is equal to twice the angle made by the straight line which bisects the remaining angle and the perpendicular from its vertex to the base. 22. From a given point in a side of a parallelogram, to draw a straight line that will bisect the parallelogram. 23. From a given point in the side of a triangle, to draw a straight line that will bisect the triangle. 24. Equal triangles between the same parallels, are upon equal bases. 25. If a straight line which is parallel to either side of a triangle bisect one of the other sides, it will also bisect the remaining side. 26. In figure to Prop. 39, if FD, GH, KE, be joined, the triangles FBD, GAH, KCE, are each equal to the triangle ABC. 27. If the square described upon one of the sides of a triangle be equal to the sum of the squares described upon the other two sides of it; the angle contained by these two sides is a right angle. 28. One side of a right-angled triangle is 543, and the other side 367; find the hypothenuse by means of Prop. 39. 29. One side of a right-angled triangle is 645, and the hypothenuse 824; find the remaining side by the same Prop. 30. If the sides of a right-angled triangle be a and b, and the hypothenuse h, show algebraically that a=(h+b) (h−b)2. 31. The sum of the 4 lines drawn from a point within a trapezium to the 4 angles, is the least, when that point is the intersection of the diameters. 32. If one angle of a triangle be equal to the other two together, it is a right angle. 33. The diameters of a rhombus bisect each other at right angles. 34. The diameters of a parallelogram bisect each other. 35. If two exterior angles of a triangle be bisected, and from the point of intersection of the bisecting lines, a line be drawn to the opposite angle of the triangle, it will bisect that angle. 36. Inscribe a square in a given rightangled isosceles triangle. 37. Inscribe a square in a given quadrant of a circle. 38. Inscribe a circle in a given quadrant of a circle. 39. If two straight lines cut each other at |