Let AB and CD (Fig. 285) be two given lines; make EH equal to A B, and HG equal to CD; then E G equals the sum of the two lines.

Make FG equal to A B, which is equal to EH.

Bisect E G in J; then, also, J bisects HF; for -

EJ=JG, and -

EH=FG.

Subtract the latter from the former; then -

EJ- EH= JG-FG;

but -

Ey-EH=HJ,

and -

JG-FG = JF;

therefore - .

HJ=JF.

Now, E J is half the sum of the two lines, and HJ is half the difference; and -

Ey-HJ=EH=AB.

Or: Half the sum of two quantities, minus half their difference, equals the smaller of the two quantities.

Let the shorter line be designated by a, and the longer by b; then the proposition is expressed by -

a =a+b/2 - b-a/2 (128.)

We also have E J + JF= EF = CD; or, half the sum of two quantities, plus half their difference, equals the larger quantity,