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 ▲ 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 bases 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 s of a ▲ a straight line be drawn through the centre of its inscribed O, 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, which are adjacent to the remaining, are in the same straight

line.

XLVII. If four straight lines intersect one another, and form four As, the Os which circumscribe them will pass through one and the same point.

XLVIII. If from the vertical < of a ▲ two lines be drawn to the base, makings with the adjacent sides, the'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 ▲ 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 2 of the line drawn from the vertex together

H

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 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 O 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

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 extremity to the given O.

LIV. If three right lines fb, ae, cd, be drawn from the sf, a, c, of a A to intersect in the same point o,

within or without the A, then ab: bc::

(Cor. 4. 6).

Sad: df

fe: ec

LV. Drawing the right line mno to cut the three sides

of the A abc, then bn : nc :

(bn: nar

CO : ao s

::

&c. (6 Def. B. 5).

Sbm: ma
Zao: dc

}

and bm ma

N. B. In the construction for Prop. 54, it may be shewn, that fox aoxoc: obxodxoe:: fax acxcf: fdx ab Xce or to daxbcxef.

LVI. In the same construction; if through ƒ 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 fwill bisect this portion.

LVII. In the same construction; if fb be at right s to ac, it will bisect the dbc. Hence the ▲ of minimum

periphery inscribed in a given A, is that formed by joining
the feet of the three perpendiculars from the <s.

LVIII. In the same construction; if de be drawn, the
right lines fb, ae and de, are all three cut harmoni-
cally.

LIX. In the same construction; if the lines ed, db, be
be produced to cut ac, cf, fa, they will be cut harmoni-
cally; as also the lines ac, cf, fa.

LX. In the same construction; the points of occurse
of cd, df, be, with ac, cf, fa, respectively, lie in directum.
N. B. There is a false demonstration of this theorem in
Bland's Geometry, see page 119.

LXI. If from a point without a O a line pc, through the
centre, and a tangent pt be drawn, and from the point of
contact a diameter td, then if any other secant pec" bet
drawn, and de' de" be joined, they will intersect portions
towards the centre, on the line pc (21 and 22. 3 El. and
Cor. 4. 6). This theorem might be proposed as a porism.

LXII. If a pair of tangents be drawn from a point
without a O, and a line joining the points of contact; any
secant from the point without is cut harmonically.

LXIII. If a third tangent be drawn intersecting the
former two and the line joining the point of contact, it
will be cut harmonically. (6. 2. El.)

LXIV. If tangents ba be be drawn to the circle abc,
and any secant pbd, and if the points a, b, c, d, be joined,
then abXcdbcxad (def. of ex æquo and 32. 3. El).

LXV. In the same construction let m be the middle
point of ac, then if any chord whatsoever omn be drawn,
the angle opm= mpn, (21st and 22d B. 3).

+

LXVI. In the same construction; if tangents be drawn
at the points n, o, they will always meet in a perpendicu- +

lar to mp at p.

LXVII. If the point p be given, and tangents be drawn
+
at the points s, n, they will meet at the perpendicular to
mp at the point m.

LXVIII. If from the occurse of two tangents any
pair of secants be drawn, the lines joining the opposite
extremities of the parts of them within the O, intersect in
the line joining the points of contact.

LXIX. A chord of a is divided in continued pro-
portion, by the straight lines inflected to any point in the
opposite circumference from the extremities of a parallel

tangent, which is limited by another tangent applied at the origin of the chord.

LXX. If from a point without a two tangents be drawn, and also any other right line to meet it in two points, the lines joining these points and one of the points of contact shall cut off segments from the chord which passes through the other point of contact parallel to the tangent.

=

LXXI. If from a point without a two lines be drawn touching the O, and from the extremities of any diameter lines be drawn to the points of contact, cutting each other within the O; the line produced, which joins their intersection and the point without the O, will be perpendicular to the diameter.

LXXII. If a be circumscribed by a A, and lines be drawn from the s of the A to the points of contact, meeting the O again in three points; tangents drawn at these points will intersect each other in the said lines produced. Also, if the said tangents and the sides of the A be produced till they meet, the three points of intersection will be in the same straight line.

LXXIII. If within a semicircle a s n on any ordinate sb another semicircle be described, and it centre o be joined with n cutting the second semicircle in t, and s t v be drawn, then ab: bv: : bv: v n.

LXXIV. Let A and B be two points without a of which O is the centre, and let BO x OC = □2 of radius, and AO × OD of rad.; and let BD and AC meet in E. Then if the chord FG be drawn' through E, and AF, FB, BG, AG be joined, AF × FB : AG × GB: :: FE EG.

LXXV. The lines from the points of contact of inscribed This, so clumsily proved in Bland, of Theorem 54.

1

angles of a A to the all meet in one point. is an evident consequence

LXXVI. AB is a chord in a given in which stand two As ACB, ACB, then let AC, BC meet in p, and AC', BC, meet in p', the line pp' produced, always passes through a given point.

This is the prize question in the Ladies' Diary for 1822, and is a plagiarism from Hamilton's Conic Sec

tions.

1

LXXVII. If the opposite sides of an irregular hexagon inscribed in a O be produced till they meet, the three points of intersection will be in the same straight line. This is the prize question of 1806, in Leybourn's Mathematical Repository; it is there solved by Earl Stanhope, Mr. Ivory, Mr. Lowry, and Mr. Nicholson. Mr. Ivory's solution, which obtained the prize, is that adopted by Bland.

LXXVIII. If common tangents be drawn to each pair of three unequal Os, the points of occurse lie in a right line.

LXXIX. If transverse tangents be drawn to two pair of these, the points in which they cut the lines joining the centres, will be in directum with the point where the direct tangents to the third pair meets the line joining their centres.

LXXX. In the same figure, the lines from the centres to the point of occurse of the transverse tangents, with the lines joining the centres, intersect in a common point.

LXXXI. The 2 of the area of a A is to the continued product of the semiperimeter, and semiperimeter diminished by each side separately, viz. A2 =S. (s—a). (s—b). (s-c), where s is the semiperimeter and a, b, c the three sides.

2

LXXXII. If the four Os be described that touch each, three sides of a A, the 2 of the area = the continued product of the four radii (By Theorem 81.)

LXXXIII. If semicircles be described on the segments of the diameter of a semicircle, and a O be described to touch those three, and other Os be described to touch two of the semicircles and this O, and so on; the perpendicular from the centre of the first its diameter, from that of the second twice its diameter, &c. and from that of nth = n times its diameter.

LXXXIV. Let A, B be two given points in the diameter of a O, equally distant from the centre C. In AB take any point P, so that PC x rad. = AC. Let CP meet the circumference in D. Draw AE, BE to any point in the circumference, and take the arc DQ double DE, join PQ; then shall AE× BE=PQ × rad.

LXXXV. If a hexagon be described about a circle, the three diagonals drawn through the opposite angular points of the figure will intersect each other in the same point.