on the same side of it not less than two right angles, these lines shall never meet on that side, if produced ever so far. For, if it be possible, let two straight lines meet, which make, with another straight line, the two interior angles, on the same side, not less than two right: Then it is plain, that the three straight lines will thus include a ▲, two of which are not less than two right angles; which (E. 17. 1.) is absurd. Wherefore, the two straight lines cannot meet, on that side of the straight line, on which they make the two interior not less 15. COR. Two straight lines, which are both perpendicular to the same straight line, are parallel to each other. PROP. IX. 16. THEOREM. The three sides of a triangle taken together, exceed the double of any one side, and are less than the double of any two sides. For, since (E. 20. 1.) any two sides of a ▲ are > the third, if the third side be added both to those two and to itself; it is evident that the three sides are, together, > the double of the third. Again, since (E. 20. 1.) any side of a ▲ is < the other two, if the other two be added both to that side, and to themselves, it is evident, that the three sides are, together,< than the double of the other two. PROP. X. 17. THEOREM. Any side of a triangle is greater than the difference between the other two sides. If, the A be equilateral, or isosceles, the proposition is manifestly true. But let it be a scalene A: Then, since (E. 20. 1.) any two sides of the ▲ are the third, if either of those two be taken from that third side, it is plain that the remaining side is greater than the difference of the other two. PROP. XI. 18. THEOREM. Any one side of a rectilineal figure is less than the aggregate of the remaining sides. Let ABCD be a given rectilineal figure: Any E F B C one side, as BC, is less than the aggregate of the remaining sides. For, first, let the figure be quadrilateral; and join B,D: Then (E. 20. 1.) BD + DC > BC; and, BA+AD>BD; ..BA + AD+DC > BD+DC; much more, then, is BA+AD+ DC > BC. And the proposition may, in the same manner, be proved to be true, when the figure has more than four sides. PROP. XII. 19. THEOREM. The two sides of a triangle are together, greater than the double of the straight line which joins the vertex and the bisection of the base. Let ABC be any given A, and let AD be the straight line joining the vertex A, and the bisection, D, of the base BC: AB+ AC> 2AD. Produce AD to E, and cut off (E. 3. 1.) DEAD; also, join B, E. Then since (hyp.) BD=DC, and (constr.) ADDE, the two sides BD, DE, of the ▲ BDE, are equal to the two sides AD, DC of the A ADC; and (E. 15. 1.) the BDE=ADC; ... (E. 4. 1.) BE=AC. But (E. 20. 1.) AB + BE > AE ; but AC has been proved to be equal to BE, and AE is (constr.) the double of AD; ... AB+ AC> 2 AD. PROP. XIII. 20. THEOREM. The two sides of a triangle are, together, greater than the double of the straight line drawn from the vertex to the base, bisecting the vertical angle. Let ABC be any given A, and let AD be drawn from the vertex A, to the base BC, bisecting the vertical BAC: Then, AB + AC>2 AD. If the given ▲ be isosceles, the straight line which bisects the vertical is (E. 4. 1.) L to the base; and since (E. 17. 1. and E. 19. 1.) each of the equal sides is greater than the perpendicular, the proposition, is, in this case, manifestly true. But, let ABC be a scalene A, and let the side AB be less than AC: Then, of the segments into which AD, bisecting the 4 BAC, divides the base BC, BD, which is adjacent to the less side AB, is the less. For, from AC, the greater, cut off (E. 3. 1.) AE AB, the less, and join D, E; and because BA, AD are equal to EA, AD, and (hyp.) the < BAD= 2 EAD, ... (E. 4. 1.) BD=DE, and < BDA = 2 EDA; but (E. 16.1.) ≤ DEC > < ADE; ... ¿DEC > 4 ADB; and (E. 16. 1.) < ADB > < ACD; much more then is ▲ DEC> < ECD; ... (E. 19. 1.) DC > DE; but it has been shewn that DE=DB; ... DC > DB. From DC, the greater cut off (E. 3. 1.) DF=DB; and join A, F: Then (E. 16. 1.) the AFC > < ABC; and because (hyp.) AC > AB, .. (E. 18. 1.) < ABC > < ACB; much more then is AFC> LACF; ... (E. 19. 1.) AC>AF: But (S. 12. 1. and constr.) AB+AF>2AD; much more then is AB+AC>2 AD. 21. COR. From the demonstration it is manifest, that of the segments into which the straight line bisecting the vertical of a scalene A, divides the base, that which is adjacent to the less side, is the less. PROP. XIV. 22. THEOREM. If a trapezium and a triangle stand upon the same base, and on the same side of it, and the one figure fall within the other, that which has the greater surface shall have the greater perimeter. |