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Ex. 2. Shew that the sum of the straight lines, joining the angles of a triangle with a point within the triangle, is less than the perimeter of the triangle, and greater than half the perimeter.
PROPOSITION XXII. PROBLEM.
To make a triangle, of which the sides shall be equal to three given straight lines, any two of which are together greater than the third.
Let A, B, C be the three given lines, any two of which are together greater than the third.
It is required to make a ▲ having its sides respectively.
Take a st. line DE of unlimited length.
In DE make DF=A, FG=B, and GH=C.
Join FK and GK.
Then KFG has its sides = A, B, C respectively.
A, B, C
..a ▲ KFG has been described as reqd.
Q. E. F.
Ex. 1. Draw an isosceles triangle having each of the equal sides double of the base.
PROPOSITION XXIII. PROBLEM.
At a given point in a given straight line, to make an angle equal to a given angle.
LA=AG, .. LA=DE ; and ·.· HL=HK, .. HL=FD.
Let A be the given pt., BC the given line, DEF the
It is reqd. to make at pt. A an angle = 4 DEF.
With centre A, and distance AG, describe O GLM.
Join AL and HL.
Then in AS LAH, DEF,
· LA=DE, and AH-EF, and HL=FD ; .. ¿ LAH= 4 DEF. .. an angle LAH has been made at pt. A as was reqd.
Q. E. F.
NOTE. We here give the proof of a theorem, necessary to the proof of Prop. XXIV. and applicable to several propositions in Book III.
PROPOSITION D. THEOREM.
Every straight line, drawn from the vertex of a triangle to the base, is less than the greater of the two sides, or than either, if they be equal.
In the ▲ ABC, let the side AC be not less than AB.
Take any pt. D in BC, and join AD.
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 must be greater than the base of the other.
.. much more then
In the As ABC, DEF,
Then must BC be greater than EF.
BAC be greater than ▲ EDF.
Of the two sides DE, DF let DE be not greater than DF.* At pt. D in st. line ED make ▲ EDG= 2 BAC,
and make DG-AC or DF, and join EG, GF. Then: AB=DE, and AC=DG, and ▲ BAC=▲ EDG,
.. BC=EG, :: DG=DF,
:. ▲ DFG= 2 DGF;
.. BC is greater than EF.
Q. E. D.
*This line was added by Simson to obviate a defect in Euclid's proof. Without this condition, three distinct cases must be discussed. With the condition, we can prove that F must lie below EG.
For since DF is not less than DE, and DG is drawn equal to DF, DG is not less than DE.
Hence by Prop. D, any line drawn from D to meet EG is less than DG, and therefore DF, being equal to DG, must extend beyond EG.
For another method of proving the Proposition, see p. 113.
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 greater than the base of the other; the angle also, contained by the sides of that which has the greater base, must be greater than the angle contained by the sides equal to them of the other.
In the As ABC, DEF,
let AB DE and AC=DF,
and let BC be greater than EF.
Then must BAC be greater than ▲ EDF.
For BAC is greater than, equal to, or less than ▲ EDF.
Now BAC cannot ▲ EDF,
for then, by 1. 4, BC would=EF; which is not the case. And BAC cannot be less than ▲ EDF,
for then, by 1. 24, BC would be less than EF; which is not the case;
.. BAC must be greater than ▲ EDF.
Q. E. D.
NOTE.-In Prop. xxvi. Euclid includes two cases, in which two triangles are equal in all respects; viz., when the following parts are equal in the two triangles :
1. Two angles and the side between them.
2. Two angles and the side opposite one of them.
Of these we have already proved the first case, in Prop. B, so that we have only the second case left, to form the subject of Prop. XXVI., which we shall prove by the method of superposition.
For Euclid's proof of Prop. XXVI., see pp. 114-115.