SUMMARY OF THE RESULTS ARRIVED AT IN EUCLID I. 27-48. SECTION II.—Parallel Straight Lines and Parallelograms. A. Theorems. 1. Two straight lines are parallel, if— (1) The alternate angles are equal (Prop. 27). (2) An exterior angle is equal to its interior opposite angle (Prop. 28). angles (Prop. 28). (4) The two lines are parallel to the same straight line (Prop. 30). (5) The two lines join the extremities of two equal and parallel straight lines towards the same parts (Prop. 33). 2. Conversely, if two straight lines are parallel — (1) The alternate angles are equal (Prop. 29). (2) An exterior angle is equal to its interior opposite angle (Prop. 29). 3. In any triangle (1) The exterior angle is equal to the sum of the two interior opposite angles (Prop. 32). (2) The three interior angles are together equal to two right angles (Prop. 32). 4. In any rectilineal figure (1) All the interior angles together with four right angles are equal to twice as many right angles as the figure has sides (Prop. 32, Cor. 1). (2) All the exterior angles are together equal to four right angles (Prop. 32, Cor. 2). 5. In any parallelogram the opposite sides and angles are equal (Prop. 34). B. Problem. To draw a straight line through a given point parallel to a given straight line (Prop. 31). SECTION III.-Areas of Parallelograms and Triangles. A. Theorems. 1. Parallelograms or Triangles on the same base or on equal bases and between the same parallels are equal (Props. 35, 36, 37, 38). 2. Conversely, equal triangles on the same base or on equal bases and on the same side are between the same parallels (Props. 39, 40). 3. A parallelogram is double a triangle on the same base and between the same parallels (Prop. 41). 4. The complements of parallelograms are equal (Prop. 43). 5. In a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the two sides (Prop. 47). 6. Conversely, if the square on one side of a triangle is equal to the sum of the squares on the other two sides, the triangle is right-angled (Prop. 48). B. Problems. 1. To describe a Parallelogram equal to a given triangle or rectilineal figure, having a given angle, and (if required) on a given straight line (Props. 42, 44, 45). 2. To describe a square on a given finite straight line (Prop. 46). ADDITIONAL PROPOSITIONS. The following Propositions are all important, although they are not included in Euclid's Text. Many of them have been already given as Exercises or Riders; but they are repeated here in order to bring together the results. As they are important, hints are given in all cases for their solution; the student will, however, do well to attempt at first unaided the solution of each of these additional propositions, and if he fails he can then make use of the hints supplied. I.-Triangles. 1. If two right-angled triangles have their hypotenuses equal, and one side equal to one side, the triangles shall be equal in all respects. [(1) Place the two triangles so that their shortest sides coincide and the right angles are adjacent; prove that two other sides are in one line by Prop. 14; that two angles are equal by Prop. 5; and that the two triangles are equal by Prop. 26; or, (2) Prove the third sides equal by Prop. 47.] 2. If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles opposite to one pair of equal sides equal, then the angles opposite to the other pair of equal sides shall be either equal or supplementary; and in the former case the triangles shall be equal in all respects. [Place the triangle ABC on the triangle DEF so that AB coincides with DE and BC falls along EF; if C coincides with F the triangles are equal in all respects; if not, the angles ACB and DFE are supplementary.] 3. The straight line which joins the middle points of two sides of a triangle is parallel to the third side, and equal to half of it (see Prop. 39, Ex. 1). 4. The straight line DE drawn through the middle point D of the side AB of the triangle ABC, parallel to the side BC, bisects the third side AC. [Draw CF parallel to AB to meet DƐ produced in F; prove the triangles ADE, CFE equal by Prop. 26.] 5. The medians of a triangle are concurrent, that is, meet at a point, and divide each other in the ratio 2: 1. [Let the medians BE, CF intersect in 0; (1) Prove, by Prop. 38 the triangles AOB, AOC, and also the triangles OBD, OCD equal; therefore AO, OD, and AD both bisect ABC; therefore they are identical; or, (2) Join AO and produce it to meet BC in X and the line BG, which is parallel to OC, in G; CO bisects AG, therefore BOE is parallel to GC and BOCG is a parallelogram, and its diagonals bisect each other.] 6. The straight lines drawn at right angles to the sides of a triangle, from their middle points, are concurrent. [Let two perpendiculars OY, OZ meet in O, join OA, OB, OC, and also OX; prove that OB=0A=OC by Prop. 4, and then by Prop. 8 that OX is at right angles to BC.] 7. The straight lines which bisect the angles of a triangle are concurrent. [Let two bisectors meet in O and join OA; draw OL, OM, ON perpendicular to the sides, and prove them equal by Prop. 26; prove AN equals AM by Prop. 47, and that the angle NAO is equal to the angle MAO by Prop. 8.] 8. The perpendiculars from the angles of a triangle on the opposite sides are concurrent. [Through A, B, C draw lines parallel to BC, CA, AB respectively; then by No. 6 the perpendiculars at A, B, C are concurrent.] 9. To bisect a triangle by a straight line drawn through a given point in one of its sides. (See Note on Additional Proposition, p. 97.) 10. To construct a right-angled triangle, having given the hypotenuse and the sum of the remaining sides. [Take a line AB equal to the sum of the sides; make the angle BAC equal to half a right angle, and with centre B and radius equal to the hypotenuse describe a circle cutting AC in D and E; draw DF and EG perpendicular to AB; DFB and EGB are each the triangle required.] 11. To construct a triangle, having given the perimeter and each of the angles at the base. [Take AB equal to the perimeter; make the angles BAP and ABP equal to half the given angles respectively; again make the angles APQ and BPR equal to half the given angles respectively, Q, R being in AB; PQR is the triangle required. Prove by Prop. 32.] 12. To construct a triangle, having given the base, the difference of the angles at the base, and the difference of the remaining sides. [Take the base AB; make the angle ABC equal to given difference of base angles; with centre A and radius equal to given difference of sides describe a circle, cutting BC in D and E, D being nearer B; join AD and produce it to P, making the angle DBP equal to PDB; PAB is the triangle required.] 13. To construct a triangle, having its three medians equal to three given straight lines. [Construct a triangle COG, having its sides equal to two-thirds of the given medians; finish the construction from proof (2) in No. 5.] II.-Lines. 14. AB is a given straight line bisected at C; show that the projections of AC, CB on any other straight line PQ are equal. [Through A draw a line parallel to PQ, meeting the perpendiculars in D and E; AD and DE are equal by No. 4 on p. 101; and the projections are equal by Prop. 34.] 15. To divide a given straight line AB into any number of equal parts. [Draw a line AC making any angle with AB; from AC cut off any part AP, and then parts PQ, QR, RS, ST all equal to AP; join BT, and draw lines through P, Q, R, S parallel to BT; in this case AB will be divided into five equal parts. Prove, by drawing through P, Q, R, S, lines parallel to AB, that the triangles are equal by Prop. 26.] 16. To trisect a given finite straight line. [As in No. 15; or describe an equilateral triangle ABC on AB ; bisect the angles CAB, CBA by the lines AO, BO, and draw OD, OE parallel to AC and BC respectively, to meet AB in D, E; prove ODE an equilateral triangle by Props. 29 and 32; and then prove DA DO DE by Prop. 29.] = = 17. To find a point in a given straight line such that the sum of its distances from two given fixed points, on the same side of the line, may be the least possible. [From one given point draw a perpendicular to the line; produce the perpendicular to an equal length on the other side of the line; join this end of the perpendicular with the second given point; the intersection of this last line and the given line is the point required; take any other point in the line and prove the sum of its distances greater by Props. 4 and 20.] 18. A and B are two fixed points and PQ a given straight line; find a point X in PQ such that the angle AXP is equal to the angle BXQ. Does it make any difference whether the points A and B are on the same or opposite sides of PQ? [Same as No. 17.] 19. Through a given point P to draw a straight line, such that the part of it intercepted between two given lines AB, AC may be bisected at P. [Draw PD parallel to AB, and in AC make AD, DE equal to one another; EP produced is the line required. Prove by Prop. 26 as in No. 4.] 20. To draw a straight line at 1ight angles to a given straight line from the extremity of it and without producing the given line, [See Ex. 4, p. 69.] |