THEOREM XXXI. The rectangle constructed on the sum and the difference of two lines, is equivalent to the difference of the squares constructed on these two lines. Let AB be the greater line, BE = BE' the less, so that AE will represent the sum of these two lines, and AE' their difference. Construct on AE as a base, and AG= AE' as an altitude, the rectangle AENG, also the square ABCD, and draw E'IF perpendicular to AB, forming the square AE'IG. G D F C A N I K E B E The two rectangles BENK, GIFD are equal, having equal bases and equal altitudes, namely, BK = AG = GI= AE', EB =E'B=IK=IF=DG; hence it follows that the rectangle AENG is equivalent to the figure DFIKBA. But this figure is the difference of the squares constructed on AB and on IK = BE' =BE. Hence, we have rectangle AENG = square AB - square BE. FOURTH BOOK. THE PROPORTIONS OF LINES AND THE AREAS OF FIGURES IN CONNECTION WITH THE CIRCLE. DEFINITIONS. I. Similar arcs, Sectors, or Segments, are those which, in different circles, correspond to equal angles at the centre. II. When two similar sectors are superposed so that their equal angles coincide, their difference is called a circular trapezoid. Thus the space BB'C'C is a circular trapezoid. III. The space included between two concentric circumferences is called a circular ring. IV. The arc of a circle is said to be rectified, when it is devel oped; that is, unfolded or drawn out into a straight line. It is only in this rectified form that we can conceive of its length, since it could not otherwise be compared with the linear unit. PROPORTIONAL LINES CONNECTED WITH THE CIRCLE. THEOREM I. When two chords intersect each other within a circle, the seg ments will be reciprocally or inversely proportional. B P E Let the two chords AA', BB' intersect at P. Drawing the auxiliary chords AB', A'B, we thus obtain two triangles PAB', PA'B, which are similar, since the angles at P are equal (B. I., T. I.); the angles at A and B are equal, so also are the angles at A' and B' (B. II., T. X.), hence these triangles are equiangular, and consequently similar (B. III., T. VI.), and their homologous sides give this proportion, PA: PB:: PB': PA'. Scholium I. When one of the chords is a diameter, and the other is perpendic ular to it, we have PA:PB:: PB:PA'. The line PB, drawn perpendicular to a diameter, is called an ordinate to this diameter. Hence, we have this condition: B' In a circle any ordinate to a diameter is a mean proportional between the two segments of this diameter. We can show that this property is one of the consequences of the properties of right-angled triangles; because if we join A, A' with the point B, we shall form a right-angled triangle (B. II., T. X., C. I.), which gives (B. III., T. XIII.) this proportion, PA: PB:: PB:PA'. Scholium II. This same right-angled triangle ABA' gives the proportion, That is, AA':AB::AB: AP. Any chord which is drawn through the extremity of a diameter is a mean proportional between its projection (B.. III., T. XIV., S. II.) on the diameter and the diameter itself. THEOREM II. P A' B' When two secants intersect each other, without a circle, they will be reciprocally proportional to their external segments. Let the two secants intersect each other at P. Draw, as in the preceding Theorem, the chords AB', A'B, and the two triangles AB'P, BA'P, will be similar, since the angle at P is common, and A=B, each being measured by half of the arc A'B' (B. II., T. X.), and the homologous sides give the proportion PA: PB::PB': PA'. A E B THEOREM III. When a secant and tangent are drawn from the same point, the tangent is a mean proportional between the secant and its external segment. This Theorem is in reality only a particular case of the preceding, when the points B, B' of the secant PB, are united in one point. But we will give a direct demonstration, by drawing the chords AB, A'B. We have, in effect, two triangles, PAB, PA'B, similar, since the angle at P is common and the two angles at A and B are equal (B. II., T. X., C. II.). Hence, we have this proportion, A A P E B PA: PB:: PB: PA'. When a straight line, as PA, is thus divided at a point A'. so that the greater segment AA', is a mean proportional between the whole line and the other segment, it is said to be divided into mean and extreme ratio. or, The above proportion will give AA': PA-AA':: PA': AA'— PA', AA': PA'::PA':AA'— PA'. Now if we take PA" equal to PA', and observe that AA'= PB, and consequently that AA'-PA'= PB-PA"= A′′B, we shall have PB: PA":: PA": A"B. From this we see that the straight line PB is also divided at the point A" into mean and extreme ratio. Scholium II. The three preceding Theorems give immediately these two equations, PA× PA'=PB × PB', PA× PA'= PB2, which may be included in one single proposition, as follows: The product of the distances from the same point, either within or without a circle, to two points of its circumference, taken in the same straight line, is always the same. In the case of the tangent we must regard the two points of the circumference as united. THEOREM IV. In a quadrilateral inscribed in a circle, the rectangle of the diagonals is equal to the sum of the rectangles of the opposite |