102 
ARCHIMEDES 
of the two smaller semicircles adjoin, and NP be drawn at 
right angles to AB meeting the external semicircle in P, the 
area of the dp/S^Xos (included between the three semicircular 
arcs) is equal to the circle on PA as diameter (Prop. 4). In 
Prop. 5 it is shown that, if a circle be described in the space 
between the arcs AP, AN and the straight line PN touching 
all three, and if a circle be similarly described in the space 
between the arcs PB, NB and the straight line PN touching 
all three, the two circles are equal. If one circle be described 
in the dp(Sr]Xos touching all three semicircles, Prop. 6 shows 
that, if the ratio of AN to NB be given, we can find the 
relation between the diameter of the circle inscribed to the 
dpfSrjXos and the straight line AB; the proof is for the parti 
cular case AN = f BN, and shows that the diameter of the 
inscribed circle = T 6 g AB. 
Prop. 8 is of interest in connexion with the problem of 
trisecting any angle. If AB be any chord of a circle with 
centre 0, and BC on AB produced be made equal to the radius, 
draw CO meeting the circle in D, E; then will the arc BD be 
one-third of the arc AE (or BF, if EF be the chord through E 
parallel to AB). The problem is by this theorem reduced to 
a reacTi? (cf. vol. i, p. 241).