Any two sides of a triangle are together greater than the third side.

B

D

Let ABC be a A.

Then any two of its sides must be together greater than the third side.

Produce BA to D, making AD=AC, and join DC.

that is, BD=BA and AC together;

.. BA and AC together are greater than BC.

Similarly it may be shewn that

AB and BC together are greater than AC,

and BC and CA

AB.

I. A.

I. 19.

Q. E. D.

Ex. 1. Prove that any three sides of a quadrilateral figure are together greater than the fourth side.

Ex. 2. Shew that any side of a triangle is greater than the difference between the other two sides.

Ex. 3. Prove that the sum of the distances of any point from the angular points of a quadrilateral is greater than half the perimeter of the quadrilateral.

Ex. 4. If one side of a triangle be bisected, the sum of the two other sides shall be more than double of the line joining the vertex and the point of bisection.

S. E.

If, from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle; these will be together less than the other sides of the triangle, but will contain a greater angle.

E

B

Let ABC be a A, and from D, a pt. in the ▲, draw st. lines to B and C.

Then will BD, DC together be less than BA, AC,

but BDC will be greater than ▲ BAC.

Produce BD to meet AC in E.

Then BA, AE are together greater than BE.

I. 20.

Add to each EC.

Then BA, AC are together greater than BE, EC.

Again, DE, EC are together greater than DC.

I. 20.

Add to each BD.

Then BE, EC are together greater than BD, DC.

And it has been shewn that BA, AC are together greater than BE, EC;

.. BA, AC are together greater than BD, DC.

Ex. 1. Upon the base AB of a triangle ABC is described a quadrilateral figure ADEB, which is entirely within the triangle. Shew that the sides AC, CB of the triangle are together greater than the sides AD, DE, EB of the quadrilateral.

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.

H

-B

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.

=

A, B, C

Take a st. line DE of unlimited length.
In DE make DF=A, FG=B, and GH=C.
With centre F and distance FD, describe o DKL.
With centre G and distance GH, describe

I. 3.

HKL.

Join FK and GK.

Then ▲ KFG has its sides =A, B, C respectively.

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.

B

H

K

M

Let A be the given pt., BC the given line, DEF the given 4.

It is reqd. to make at pt. A an angle
= L DEF.
In ED, EF take any pts. D. F; and join DF.
In AB, produced if necessary, make AG=DE.
In AC, produced if necessary, make AH=EF.
In HC, produced if necessary, make HK=FD.

With centre A, and distance AG, describe O GLM.
With eentre H, and distance HK, describe o LKM.

Join AL and HL.

Then・・ LA=AG, .. LA=DE ;

and. HL=HK, .. HL=FD.

Ax. 1.

Ax. 1.

Then in As LAH, DEF,

·· LA=DE, and AH-EF, and HL=FD ;

.. ▲ LAH = L DEF.

I. c.

Q. E. F.

.. an angle LAH has been made at pt. A as was reqd.

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.

B

In the ▲ ABC, let the side AC be not less than AB.

Take any pt. D in BC, and join AD.

Then must AD be less than AC.

For AC is not less than AB;

.. ABD is not less than 4 ACD. But ADC is greater than ▲ ABD ;

.. ADC is greater than 4 ACD;

.. AC is greater than AD.

I. A. and 18.

I. 16.

I. 19.

Q. E. D.