The Public School Euclid and Algebra . et A C in ^.1 and A£ are together greater than BE. Prop. 20. Add to each BA and AC are together greater than BF and , DE and EC are together gi-eater than DC. Prop. 20. Add to each BE and EC are together greater than BD and it has been shown that BA and AC are together greater than BE and EC;.-. BA and AC are together greater than BD and , because Z BDC is greater than / DEC, Prop. Z DEC is greater than Z BAC, Prop. 16. •. Z ^^C is greater than Z /ii4C EXERCISES. 1. lu the figure Prop. 21, join DA, a
The Public School Euclid and Algebra . et A C in ^.1 and A£ are together greater than BE. Prop. 20. Add to each BA and AC are together greater than BF and , DE and EC are together gi-eater than DC. Prop. 20. Add to each BE and EC are together greater than BD and it has been shown that BA and AC are together greater than BE and EC;.-. BA and AC are together greater than BD and , because Z BDC is greater than / DEC, Prop. Z DEC is greater than Z BAC, Prop. 16. •. Z ^^C is greater than Z /ii4C EXERCISES. 1. lu the figure Prop. 21, join DA, and show that the sum of DA, DB and DC is less than the sum of the sides of the triangle ABC,but greater than half the sum. 2. In the figure of Prop. 21, show that the angle BDC is greater than the angle BAC, by joining AD and producing it towards the base. 42 EUCLID S ELEMENTS. PROPOSITION 22. make a triangle the sides ofwluch shall he equal to three givenstraight Ihies, any two of which are greater than the third- A B C. Let A, B, C he the three given straight lines, any two ofwhich are greater than the third. It is required to make a triangle the sides of which shall berespectively equal to A, B and C. Take a straight line DE terminated at i), but unlimitedtowards E. From it cut off DF = A, FG = B, GE = C. Prop. centre F and radius FD, describe the 0 centre G and radius GH, describe the 0 HKL,cutting the other circle at K. Join KF, is the triangle required. Because FK = FD, Def. of 0. .-. FK = GK = Gil, Def of 0. .-. GK = aAnd FG was made equal to A KFG has its sides respectively equal to A, B and C. Questions on Prop. 22. 1. Why does the enunciation state that any two of the given lines are together greater than the third ? 2. Will the circumferences of two circles intersect, if the sum of their radii is greater than the distance between their centres ? Euclids elements. 43 PROPOSITION 23. a given point in a giv
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