s j & B 46 & 8 8 84 IX. The centre of a 0, which touches the semicircles described on the two sides of a right angled 4, is in the middle point of the hypothenuse. X. If on three sides of a right angled a semicircles be described, and with the centres of those described on the sides, Os be described touching that described on the base ; they will also touch the other semicircles. XI. If two Os touch one another internally at a point t, and a line be drawn intersecting the outer in the points a, a' and the inner in the points b, b, the atb = <b't á (32. 3. Elr.) XII. The sum of the O's of the lines from any point of a 0 to the es of the inscribed 2 is = to eight times the O2 of the radius. And if drawn to the 28 of the inscribed equilateral a they are equal to six times the rad.2 XIII. Any two radii of a O being drawn, and any point taken in the intercepted arc, and perpendiculars being let fall on the radii from said point; the line joining the extremities of these perpendiculars is = to the perpendicular from the extremity of one of the given radii on the other. This theorem is of use in trigonometry. XIV. Let mon be the perpendicular diameter of the circumscribed O, and bv a perpendicular of the A produced to the circumference, and vs a perpendicular from the point in which this last meets the circumference on the perpendicular diameter, then on Xns = the O2 of half the difference of the sides. XV. If two Os touch one another, and also touch a right line; the part of the line between the points of contact is a mean proportional between the diameters of the Os. XVI. If the opposite sides of a quadrilateral figure inscribed in a o be produced, the 0 of the line joining the occurses = the sum of the O’rs the tangents from the same points. XVII. When any number of Os pass through the same two points, tangents drawn from any point in either of the two external circumferences to the remaining Os shall be to one another in a given ratio. XVIII. If from any point in the periphery of a 0 perpendiculars be let fall on the three sides of the inscribed A; the feet of those perpendiculars lie in directum. XIX. P is a point in the diameter of a 0, and PB a perpendicular to the diameter; draw any chord PA and a tangent AB meeting PB in B, and draw BD and CE (C the centre) perpendicular to P A; then PE=DÀ. XX. Within a given A suppose another A to be inscribed, by joining the middle point of its sides; and again within this A suppose another to be inscribed by joining the middle point of its sides, and so on ad infinitum, what will be the limit of the aggregate, of the sum of the o’s of the sides of all the As so formed ? XXI. If the circumference of a O be divided into any number of = parts, and perpendiculars from the points be let fall on any diameter; the sum of the perpendiculars on one side of the diameter is = to the sum of those on the other. XXII. Prove that if from the extremities of the side of a pentagon inscribed in a straight lines be drawn to the middle of the arc subtended by the adjacent side, their difference is = to the radius; the sum of their o's is three times the o’ of the radius; and the rectangle contained by them is = to the o of the radius. XXIII. If four Os touch, either internally or externally three sides of any quadrilateral, their centres are always on the same circumference. XXIV. If the sides of a circumscribed quadrilateral to a O touch at the <s of an inscribed one, their diagonals cut in the same point. XXV. The diagonals of a quadrilateral in a 0, are as the sums of the rectangles of the sides that terminate in them. XXVI. The C's of the side of an equilateral A in a 0, is = the area of a regular dodecagon in the same. XXVI. The of the side of an equilateral A 3 o of radius of circumscribing (12. 2. El.) XXVIII. The o of the side of a regular pentagon in a O = the sum of the a's of the sides of a regular hexagon and decagon, in the same O. XXIX. Taking a point in the periphery of a O between any two <s of an inscribed equilateral A; the sum of the lines to it from the adjacent zs is = to the line from the opposite <. (Cor. 16. 6. Elr.) XXX. The solid under three sides of a 4 divided by four times the area, is = to the radius of the circumscribing O. (Cor. 16. 6. Elr.) XXXI. If equilateral As be described on the three sides of a A, and a, a', a' be the centres of their inscribed Os, the A a a' a formed by joining those points is an equilateral A. XXXII. The distance between the centres of the inscribed and circumscribing Os to a 4, is a mean proportional between the radius of the circumscribing and the difference between that radius and the diameter of the inscribed. (Cor. 6. 2, and 4 and 16 B. 6). XXXIII. Draw the tangents a m, bn at the ends of the diameter of a semicircle, and any tangent mpn crossing them, join a n, bm intersecting in 0, join po and produce it to the diameter at v; then pov is at right <s to the diameter, and bisected at o. XXXIV. If through the middle point of any chord of a two chords be drawn ; the lines joining their extremities will intersect the first chord at = distances from the middle point. XXXV. Let F E, the diameter of a o, described about any plane A ACB, bisect the base AB perpendicularly in H and let EP, FS be drawn perpendicular to AC; with the radii CH, CS and centre C describe Os CQG and CSD meeting AC in G and S; and let S be the nearest point to C, then from P and G draw the tangents PQ, GD and the rectangle PQXGD will be to the area of the A ACB. XXXVI. In any plane A, the difference between the O of half the sum of the sides and the o? of the line bisecting the base, multiplied into the difference between the O2 of the line bisecting the base and o’ of half the difference of the sides, is = to the o2 of the area of the A. XXXVII. The sum of the three radii of the Os that --touch one side, and the other two produced of a A is = to four times the radius of the circumscribing + the radius of the inscribed O. XXXVIII. If equilateral As be described on the three sides of a A, the lines joining their vertices with the opposite <s of the A, all pass through one point. XXXIX. If a O touch two given Os, the line joining the points of contact always passes through a given point. XL. If at the <s of a A in a tangents be drawn, they will meet the opposite sides in three points that lie in directum. XLI. If the opposite sides of a quadrilateral in a be produced to meet, and tangents be drawn at the ends of the diagonals, and produced to meet, the four points so found lie in one right line. XLII. In any plane a ABC, if lines BF,CD, be drawn from two of the <s to the opposite sides, o being their / point of intersection, and DF being joined, then will ABC: A OBC:: A ADF:A ODF. XLIII. If two Os be described to touch an ordinate of a semicircle, the semicircle itself and the semicircles on the segments of the diameter, they will be =. XLIV. A perpendicular from one of the base 4s of a A on the bisector of the vertical <, divides it into segments, which are to each other as the sum of the sides to their difference. XLV. The common secants to each two of three unequal intersecting Os meet in a point. XLVI. If from one of the 2s of a A a straight line be drawn through the centre of its inscribed 0, and a perpendicular be drawn to this line from one of the other <s; the point of intersection of the perpendicular and the two points of contact of the inscribed o, which are adjacent to the remaining C, are in the same straight line. XLVII. If four straight lines intersect one another, and form four As, the O3 which circumscribe them will pass through one and the same point. XLVIII. If from the vertical < of a A two lines be drawn to the base, making = <s with the adjacent sides, the O’s of those sides will be proportional to the rectangles contained by the adjacent segments of the base. XLIX. If from the vertex of a A there be drawn a line to any point in the base, from which point lines are drawn parallel to the sides ; the sum of the rectangles of each side, and its segments adjacent to the vertex, will be =to the Q2 of the line drawn from the vertex together 重 with the rectangle contained by the segments of the base. L. If two lines from the ends of the chords of a given segment of a 0 always cut at a given <; the line join+ ing the points where they cut the segment is always of a given length. LI. If from the extremities of the diameter of a any number of chords be drawn, two and two, intersecting each other in a perpendicular to that diameter, the lines joining the extremities of every corresponding two, will meet the diameter produced, in the same point. LII. If any number of circles cut one another in the same points, and from one of these points any number of lines be drawn, the parts of these which are intercepted between the several circumferences have the same ratio. LIII. If from the centre of a O a line be drawn perpendicular to another which is given in position, and from their point of intersection a line set off along the said perpendicular = in length to a tangent to the O from the said point of intersection, it will give a point thereon, from which if a line be drawn to that first given in position, it will be = to a tangent drawn from its ex. tremity to the given O. LIV. If three right lines fb, ae, cd, be drawn from the csf, d, c, of a Ğ to intersect in the same point o, within or without the A, then ab : bc : : Sad : df fe : ec (Cor. 4. 6). LV. Drawing the right line mno to cut the three sides of the A abc, then bn : nc :: Som : ma? and bm : + {ão : de co: ao . N. B. In the construction for Prop. 54, it may be shewn, that fo x ao Xoc : ob X od X oe :: fax acX cf : fdx ab X ce or to da Xbcx ef. LVI. In the same construction; if through f a parallel be drawn to ac, and the lines joining b with e and d be produced to intercept a portion of this parallel, the point Ŝ will bisect this portion. LVII. In the same construction; if fo be at right <s to ac, it will bisect the dbc. Hence the A of minimum Son : no&c. (6 Def. B. 5). |