that it is not equal to AB; therefore AC is greater than AB. Wherefore the greater angle, &c. Q.E.D. NY two fides of a triangle are together greater than A the third side. Let ABC be a triangle; any two sides of it together are greater than the third side, viz. the fides BA, AC greater than the fide BC; and AB, BC greater than AC ; and BC, CA greater than AB. Produce BA to the point D, - and make * AD equal to AC; - ID Because DA is equal to AC, the angle ADC is likewise equal * to ACD; but the angle BCD is greater than the angle ACD; therefore the angle BCD is great- B C er than the angle ADC; and be- cause the angle BCD of the triangle DCB is greater than its angle BDC, and that the greater “fide is opposite to the greater angle; therefore the fide DB is greater than the fide BC ; but DB is equal to BA and AC 3 therefore the fides BA, AC are greater than BC. In the same manner it may be demonstrated, that the sides AB, BC are greater than CA, and BC, CA greater than AB. Therefore any two fides, &c. Q. E. D. F, from the ends of the fide 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 tris, angle, but shall contain a greater angle. Let the two straight lines BD, CD be drawn from B, C, the ends of the fide BC of the triangle ABC, to the point D AC of the triangle, but contain an angle BDC greater than the greater than the third fide, the two fides BA, AE of the tri5 angle * er than CD, add DB to each of these ; therefore the fides CE, EB are greater than CD, DB; but it has been shewn that B BA, AC are greater than BE, EC; much more then are BA, AC greater than BD, DC. Again, because the exterior angle of a triangle is greater than the anterior and opposite angle, the exterior angle BDC of the triangle CDE is greater than CED; for the same reason, the exterior angle CEB of the triangle ABE is greater than BAC; and it has been demonstrated that the angle BDC is greater than the angle CEB ; much more then is the angle BDC greater than the angle BAC. Therefore, if from the ends of, &c. Q. E. D. P R O P. xxii. P R o B. To make a triangle of which the fides shall be equal *N* to three given straight lines; but any two whatever of these must be greater than the third a. a 22. 1. Let A, B, C be the three given straight lines, of which any two whatever are greater than the third, viz. A and B greater than C; A and C greater than B; and B and C than A. It is required to make a triangle of which the fides 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 to B, and GH equal to C; and from the centre F, at the distance FD, describe * T 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 als angle at the given point A in the given straight line A.o. .*. C A. equal to the given rectilineal angle DCE. Takein CD, CE, any points D, E, and join DE 3 and make * the the angle FAG is made equal to the given rectilineal angie DCE. Which was to be done. f - IF two triangles have two fides of the one equal to two fides of the other, each to each, but the angle contained by the two fides of one of them greater than the angle contained by the two fides 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 fides F two triangles have two fides of the one equal to two fides of the other, each to each, but the base of the one greater than the base of the other; the angle also contained by the fides of that which has the greater base, hall be greater than the angle contained by the fides equal to them, of the other. F two triangles have two angles of one equal to two angles of the other, each to each ; and one fide equal to one fide, viz. either the sides adjacent to the equal angles, or the fides opposite to equal angles in each; then shall the other sides be equal, each to each ; and also the third angle of the one to the third angle of the other. |