APPENDIX I. Enunciations of the Propositions of EUCLID, BOOK II. 1. If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two lines is equal to the sum of the rectangles contained by the undivided line and the several parts of the divided line. 2. If a straight line be divided into any two parts, the square on the whole line shall be equal to the sum of the rectangles contained by the whole line and each of the parts. 3. If a straight line be divided into any two parts, the rectangle contained by the whole line and one of the parts shall be equal to the square on that part, together with the rectangle contained by the parts. 4. If a straight line be divided into any two parts, the square on the whole line shall be equal to the sum of the squares on the parts, together with twice the rectangle contained by the parts. 5. If a straight line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of section, shall be equal to the square on half the line. 6. If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus produced and the part of it produced, together with the square on half the line bisected, shall be equal to the square on the line made up of the half and the part produced. 7. If a straight line be divided into any two parts, the square on the whole line, together with the square on one of the parts, shall be equal to twice the rectangle contained by the whole line and that part, together with the square on the other part. Enunciations of Euclid, Book II. 181 8. If a straight line be divided into any two parts, four times the rectangle contained by the whole line and one of the parts, together with the square on the other part, shall be equal to the square on the line made up of the whole and the first part. 9. If a straight line be divided into two equal parts, and also into two unequal parts, the sum of the squares on the two unequal parts shall be double the sum of the squares on half the line and on the line between the points of section. 10. If a straight line be bisected and produced to any point, the sum of the squares on the whole line thus produced, and on the part of it produced, shall be double the sum of the squares on half the line bisected, and on the line made up of the half and the part produced. 11. To divide a straight line into two parts so that the rectangle contained by the whole line and one part shall be equal to the square on the other part. 12. In an obtuse-angled triangle, the square on the side subtending the obtuse angle is greater than the sum of the squares on the sides containing the obtuse angle by twice the rectangle contained by either of these sides, and the produced part of it intercepted between the perpendicular let fall on it from the opposite vertex and the obtuse angle. 13. In every triangle, the square on the side subtending an acute angle is less than the sum of the squares on the sides containing that angle, by twice the rectangle contained by either of these sides and the part of it intercepted between the perpendicular let fall on it from the opposite vertex and the acute angle. 14. To describe a square that shall be equal to a given recti lineal figure. Enunciations of Standard Theorems and Loci. $ 31, Ex. 1. If A, B, C, D be four points in order on the same straight line, the rectangle AC BD shall be equal to the sum of the rectangles AD BC and AB CD. $ 31, Ex. 3. The square on any straight line is four times the square on half the line. § 32, Ex. 1. In a right-angled triangle, the square on the altitude from the right angle is equal to the rectangle contained by the segments into which the altitude divides the hypotenuse; and the square on either of the sides which contain the right angle is equal to the rectangle contained by the hypotenuse and its adjacent segment. $ 33, Ex. 1. The rectangle contained by the sum and the difference of two straight lines is equal to the difference between their squares. $ 33, Ex. 4. The difference between the squares on two sides of a triangle is equal to twice the rectangle contained by the base and the segment of the base intercepted between its midpoint and the foot of the altitude. $ 34, Ex. 1. If, from the vertex of an isosceles triangle, a straight line be drawn to meet the base, or base produced, the difference of the squares on this line, and on one of the equal sides of the triangle, is equal to the rectangle contained by the segments of the base. § 34, Ex. 2. If a straight line be divided externally, the square on the straight line is equal to the sum of the squares of the segments diminished by twice the rectangle contained by the segments. $ 35, Ex. 2. The square on the sum of two straight lines, together with the square on their difference, is equal to twice the sum of the squares on the two straight lines. Enunciations of Standard Theorems, etc. 183 § 37, Ex. 1. In any triangle, the sum of the squares on two sides is equal to twice the square on half the third side, together with twice the square on the median which bisects it. $ 37, Ex. 11. G is the point of intersection of the medians of the A ABC, and P is any other point; show that PA2+PB2+PC2=GA2+GB2+GC2+3PG?. $ 38, Ex. 1. If there be two straight lines, the rectangle contained by the first and the projection on it of the second is equal to the rectangle contained by the second and the projection on it of the first. § 41, Ex. 1. Find the locus of a point, the sum of the squares of whose distances from two given points is equal to a given square. $ 41, Ex. 2. Find the locus of a point, the difference of the squares of whose distances from two given points is equal to a given square. 1. A and B are two fixed points, M is the mid-point of AB, and P is any other point in the same straight line; show that when P is in AB, PA-PB=2PM, and that when P is in AB produced, PA+PB=2PM. 2. OA and OB are two fixed straight lines, OM is the bisector of the ZAOB, and OP is any other straight line through O; show that when OP falls within the ZAOB, POA_ZPOB=2 POM, and that when OP falls without the ZAOB, ZPOA+ZPOB=2_POM. 3. ABC is a triangle in which AB=AC, D is a point in the bisector of LA, DB and DC are joined and produced to E and F, so that BE=CF; prove that A AEF is isosceles. 4. ABC is a triangle whose base BC is trisected in D and E; if AD=AE, show that AB=AC. 5. In the quadrilateral ABCD, AB=CD and AC=BD. If AB and CD meet in O, show that OA=OD. 6. In A ABC, AB=AC and the straight line PQ meets AB in P, and AC produced in Q; show that Z APQ >LAQP. 7. In any quadrilateral ABCD the sum of the exterior angles at A and C is greater than either of the interior angles at Bor D. |