6. Inscribe a square in a given square, having its diagonal equal to a given straight line. With O as centre and radius= given diagonal, Join OP, and produce PO to meet BC in R. Consider the case in which P lies on AD produced. 7. Inscribe a square in a quadrant of a circle. R Bisect BAC, etc. 8. Inscribe a square in a right-angled isosceles triangle, one side being on the hypotenuse. Trisect BC in P, Q, etc. P R 9. Divide a triangle by two straight lines into three parts which can be arranged so as to form a parallelogram with a given angle. 10. A, B are two points on the same side of a straight line CD; find a point P in CD such that AP+BP may be a minimum. Use § 21, Ex. 1, and compare § 27, Ex. 1. 11. Inscribe an equilateral triangle in a square, so that a vertex of the triangle may coincide with an angular point of the square. Make an angle with the diagonal equal to half the angle of an equilateral triangle. 12. Inscribe an equilateral triangle in a square, so that a vertex of the triangle may coincide with the mid-point of a side of the square. The sides of the triangle are equal to the sides of the square. 13. In a given square inscribe a square having its sides equal to a given straight line. Find diagonal of required square, and see Ex. 6. 14. Given the base and area of a triangle, find its vertex so that its perimeter may be a minimum. See § 21, Ex. 3. 15. Given the base and perimeter of a triangle, find its vertex so that its area may be a maximum. See § 21, Ex. 8. APPENDIX. Enunciations of the Propositions and Corollaries of 1. To construct an equilateral triangle on a given straight line. 2. From a given point to draw a straight line equal to a given straight line. 3. From the greater of two given straight lines to cut off a part equal to the less. 4. If two sides and the contained angle of one triangle be respectively equal to two sides and the contained angle of another triangle, the two triangles shall be equal in every respect. 5. The angles at the base of an isosceles triangle are equal; and, if the equal sides be produced, the angles on the other side of the base shall also be equal. Cor.-Every equilateral triangle is also equiangular. 6. If two angles of a triangle be equal, the sides opposite to them shall also be equal. Cor.-Every equiangular triangle is also equilateral. 7. On the same base, and on the same side of it, there cannot be two triangles having the sides which are terminated at one end of the base equal and also those which are terminated at the other end. 8. If three sides of one triangle be respectively equal to three sides of another triangle, the two triangles shall be equal in every respect. 9. To bisect a given angle. 10. To bisect a given straight line. 11. To draw a straight line at right angles to a given straight line from a given point in the line. 12. To draw a straight line perpendicular to a given straight line from a given point outside the line. 135 13. The angles which one straight line makes with another on one side of it are together equal to two right angles. Cor. i.-If two straight lines cut one another, the angles formed shall be together equal to four right angles. Cor. ii.-If any number of straight lines meet in a point, the angles formed shall be together equal to four right angles. 14. If at a point in a straight line two other straight lines on opposite sides of it make adjacent angles which are together equal to two right angles, these two straight lines shall be in one and the same straight line. 15. If two straight lines cut one another, the vertically opposite angles shall be equal. 16. If one side of a triangle be produced, the exterior angle thus formed shall be greater than either of the two interior opposite angles. 17. Any two angles of a triangle are together less than two right angles. 18. If one side of a triangle be greater than another side, the angle opposite to the greater side shall be greater than the angle opposite to the other. 19. If one angle of a triangle be greater than another, the side opposite to the greater angle shall be greater than the side opposite to the other. 20. Any two sides of a triangle are together greater than the third side. 21. If from the ends of a side of a triangle two straight lines be drawn to a point within the triangle, these two straight lines shall be together less than the other two sides of the triangle, but they shall contain a greater angle. 22. To construct a triangle whose sides shall be equal to three given straight lines, any two of which are greater than the third. 23. At a given point in a given straight line, to make an angle equal to a given angle. 24. If two triangles have two sides of the one respectively equal to two sides of the other, but the contained angles unequal, the base of the triangle which has the greater contained angle shall be greater than the base of the other. Enunciations of Euclid, Book I. 137 25. If two triangles have two sides of the one respectively equal to two sides of the other but the bases unequal, the angle contained by the two sides of the triangle which has the greater base shall be greater than the angle contained by the two sides of the other. 26. If two angles and a side in one triangle be respectively equal to two angles and the corresponding side in another, the two triangles shall be equal in every respect. 27. If a straight line cutting two other straight lines make the alternate angles equal, the two straight lines shall be parallel. 28. If a straight line cutting two other straight lines make an exterior angle equal to the interior opposite angle on the same side of the cutting line, or make the two interior angles on the same side of the cutting line together equal to two right angles, the two straight lines shall be parallel. 29. If a straight line cut two parallel straight lines, it shall make the alternate angles equal, and any exterior angle equal to the interior opposite angle on the same side of the cutting line, and the two interior angles on the same side of the cutting line together equal to two right angles. 30. Straight lines which are parallel to the same straight line are parallel to each other. 31. Through a given point, to draw a straight line parallel to a given straight line. 32. If one side of a triangle be produced, the exterior angle shall be equal to the sum of the two interior opposite angles, and the three interior angles are together equal to two right angles. Cor. i--All the interior angles of any rectilineal figure together with four right angles are equal to twice as many right angles as the figure has sides. Cor. ii. All the exterior angles of any rectilineal figure are together equal to four right angles. 33. The straight lines which join the ends of two equal and parallel straight lines towards the same parts are themselves equal and parallel. 34. The opposite sides and opposite angles of a parallelogram are equal, and either diagonal bisects the parallelogram. 35. Parallelograms on the same base and between the same parallels are equal. |