therefore AC is not less than AB. And it has been shewn that AC is not equal to AB. Wherefore, the greater angle &c. Q.E.D. PROPOSITION 20. THEOREM. Any two sides of a triangle are together greater than the third side. Let ABC be a triangle: any two sides of it are together greater than the third side; namely, BA, AC greater than Produce BA to D, making AD equal to AC, and join DC. Then, because AD is equal to AC, the angle ADC is equal to the angle ACD. [I. 3. [Construction. [I. 5. But the angle BCD is greater than the angle ACD. [Ax. 9. Therefore the angle BCD is greater than the angle BDC. And because the angle BCD_of_the_triangle BCD is greater than its angle BDC, and that the greater angle is subtended by the greater side; therefore the side BD is greater than the side BC. But BD is equal to BA and AC. Therefore BA, AC are greater than BC. [I. 19. In the same manner it may be shewn that AB, BC are greater than AC, and BC, CĂ greater than AB. Wherefore, any two sides &c. Q.E.D. If from the ends of the side of a triangle there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle. E Let ABC be a triangle, and from the points B, C, the ends of the side BC, let the two straight lines BD, CD be drawn to the point D within the triangle: BD, DC shall be less than the other two sides BA, AC of the triangle, but shall contain an angle BDC greater than the angle BAC. B Produce BD to meet AC at E. Because two sides of a triangle are greater than the third side, the two sides BA, AE of the triangle ABE are greater than the side BE. To each of these add EC. Therefore BA, AC are greater than BE, EC. [I. 20. Again; the two sides CE, ED of the triangle CED are greater than the third side CD. To each of these add DB. Therefore CE, EB are greater than CD, DB. [I. 20. But it has been shewn that BA, AC are greater than BE, EC; much more then are BA, AC greater than BD, DC. Again, because the exterior angle of any triangle is greater than the interior opposite angle, the exterior angle BDC of the triangle CDE is greater than the angle CED. [I. 16. For the same reason, the exterior angle CEB of the triangle ABE is greater than the angle BAE. But it has been shewn that the angle BDC is greater than the angle CEB; much more then is the angle BDC greater than the angle ВАС. Wherefore, if from the ends &c. Q.E.D. PROPOSITION 22. PROBLEM. To make a triangle of which the sides shall be equal to three given straight lines, but any two whatever of these must be greater than the third. Let A, B, C be the three given straight lines, of which any two whatever are greater than the third; namely, A and B greater than C; A and C greater than B; and B and C greater than A: it is required to make a triangle of which the sides shall be equal to A, B, C, each to each. Take a straight line DE terminated at the point D, but unlimited towards E, and make DF equal to A, FG equal to B, and GH equal to C. [I. 3. D H A B C From the centre G, at the distance GH, describe the circle HLK, cutting the former circle at K. Join KF, KG. The triangle KFG shall have its sides equal to the three straight lines A, B, C. Because the point F is the centre of the circle DKL, FD is equal to FK. But FD is equal to A. [Definition 15. [Construction. Therefore FK is equal to A. [Axiom 1. Again, because the point G is the centre of the circle HLK, GH is equal to GK. But GH is equal to C. Therefore GK is equal to C. And FG is equal to B. [Definition 15. [Construction. [Axiom 1. [Construction. Therefore the three straight lines KF, FG, GK are equal to the three A, B, C. Wherefore the triangle KFG has its three sides KF, FG, GK equal to the three given straight lines A, B, C. Q.E.F. PROPOSITION 23. PROBLEM. At a given point in a given straight line, to make a rectilineal angle equal to a given rectilineal angle. Let AB be the given straight line, and A the given point in it, and DCE the given rectilineal angle: it is required to make at the given point A, in the given straight line AB, an angle equal to the given rectilineal angle DCE. In CD, CE take any points D, E, and join DE. Make the triangle AFG the sides of which shall be equal to the three straight lines CD, DE, EC; so that AF shall be equal to CD, AG to CE, and FG to DE. [I. 22. The angle FAG shall be equal to the angle DCE. B Because FA, AG are equal to DC, CE, each to each, and the base FG equal to the base DE; [Construction. therefore the angle FAG is equal to the angle DCE. [I. 8. Wherefore at the given point A in the given straight line AB, the angle FAG has been made equal to the given rectilineal angle DCE. Q.E.F. PROPOSITION 24. THEOREM. If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other, the base of that which has the greater angle shall be greater than the base of the other. Let ABC, DEF be two triangles, which have the two sides AB, AC, equal to the two sides DE, DF, each to each, namely, AB to DE, and AC to DF, but the angle BAC greater than the angle EDF: the base BC shall be the two sides BA, AC are equal to the two sides ED, DG, each to each; and the angle BAC is equal to the angle EDG; [Constr. therefore the base BC is equal to the base EG. And because DG is equal to DF, the angle DGF is equal to the angle DFG. [I. 4. [Construction. [I. 5. But the angle DGF is greater than the angle EGF. [Ax. 9. Therefore the angle DFG is greater than the angle EGF. Much more then is the angle EFG greater than the angle EGF. [Axiom 9. And because the angle EFG of the triangle EFG is greater than its angle EGF, and that the greater angle is subtended by the greater side, therefore the side EG is greater than the side EF. But EG was shewn to be equal to BC; therefore BC is greater than EF. Wherefore, if two triangles &c. Q.E.D. [I. 19. PROPOSITION 25. THEOREM. If two triangles have two sides of the one equal to two sides of the other, each to each, but the base of the one |