If semicircles be described on supplemental chords of a semicircle, the sum of the areas of the two crescents thus formed is equal to the area of the triangle whose sides are the supplemental chords and the diameter. PROP. XXXII.—Theorem. If two triangles (ABC, CDE) which have two sides of one proportional to two sides of the other (AB : BC :: CD : DE), and the contained angles (B, D) equal, be joined at an angle (C), so as to have their homologous sides parallel, the remaining sides are in the same right line. Dem.—Because the triangles ABC, CDE have the angles B and D equal, and the sides about these angles proportional, viz., AB : BC :: CD : DE, they are equiangular [vi.]; therefore the angle BAC is equal to DCE. To each add ACD, and we have the sum of the angles BAC, ACD equal to the sum of DCE and ACD; but the sum of BAC, ACD is [I. xxix.] two right angles; therefore the sum of DCE and ACD is two right angles. Hence [I. xiv.] AC, CE are in the same right line. PROP. XXXIII.–Theorem. In equal circles, angles (BOC, EPF) at the centres or (BAC, EDF) at the circumferences have the same ratio to one another as the arcs (BC, EF) on which they stand, and so also have the sectors (BOC, EPF). Dem.—1. Take any number of arcs CG, GH in the first circle, each equal to BC. Join OG, OH, and in the second circle take any number of arcs FI, IJ, each equal to EF. Join IP, JP. Then because the arcs BC, CG, GH are all equal, the angles BOC, COG, GOH, are all equal [III. xxvii.]. Therefore the arc BH and the angle BOH are equimultiples of the arc BC and the angle BOC. In like manner it may be proved that the arc EJ and the angle EPJ are equimultiples of the arc EF and the angle EPF. Again, since the circles are equal, it is evident that the angle BOH is greater than, equal to, or less than the angle EPJ, according as the arc BH is greater than, equal to, or less than the arc EJ. Now we have four magnitudes, namely, the arc BC, the arc EF, the angle BOC, and the angle EPF; and we have taken equimultiples of the first and third, namely, the arc BH, the angle BOH, and other equimultiples of the second and fourth, namely, the arc EJ and the angle EPJ, and we have proved that, according as the multiple of the first is greater than, equal to, or less than the multiple of the second, the multiple of the third is greater than, equal to, or less than the multiple of the fourth. Hence [V. Def. v.] BC : EF :: the angle BOC : EPF. Again, since the angle BAC is half the angle BOC [III. xx.], and EDF is half the angle EPF, BAC : EDF :: BOC : EPF [V. xv.]; but BOC : EPF :: BC : EF. Hence BAC : EDF :: BC : EF [V. xi.]. 2. The sector BOC : sector EPF :: BC : EF.