LXXXVIII. Suppose a quadrilateral figure be inscribed in another quadrilateral figure, so that the perimeter of the inscribed one be a minimum; then the opposite sides of the inscribed figure, when produced, will intersect each other in the diagonals produced of the circumscribed figure.

LXXXIX. Let r r " be the radii of the three Os that touch, each one side of a ▲ and the other two produced; then rxr+rxr" + rxr" = 2 s2, where s is the semiperimeter.

XC. If a be inscribed in a ▲, the rectangle under the segments of the base, made by the point of contact, is = to the rectangle under the perpendiculars from the ends of the base on the bisector of the vertical .

XCI. The centre of the inscribed in a ▲, the point of contact with its base, and the middle point of the line from the vertex to the point of contact, are in di

rectum.

XCII. Let AB be the side of a hexagon, inscribed in the DAB, and let ABC be an equilateral ▲, described on AB, so that the C is outside the O; let AD and DB be right lines inflected to any point in the circumference, and let CE be drawn parallel to AD, meeting BD in E; the 's of AD and CE are together the of BD and twice the 2 of AB, the radius of the Q.

2

to

XCIII. If, from any number of given points A, A', A", &c. situated in a right line, outside a given O, tangents AB, A'B', A"B", &c. be drawn; and if, from the centres A, A', A", &c. and with the radii AB, A'B', A"B", &c. os be described, they will intersect one another in the same point.

XCIV. If, from any point in the circumference of a O, right lines be drawn to the extremities of a chord, and meeting the perpendicular diameter, they will divide that diameter internally and externally in the same ratio.

XCV. If two right lines be inflected from the extremities of the base of a ▲ to cut the opposite sides proportionally, another right line, drawn from the vertex through the point of contact, will bisect the base.

XCVI. If a semicircle be described on the side of a rectangle, and through its extremities two right lines be drawn from any point in the circumference, to meet the opposite side produced both ways, the altitude of the rect

angle will be a mean proportional between the segments thus intercepted.

XCVII. If through any point in the circumference of a O, two right lines be drawn parallel to adjacent sides of an inscribed quadrilateral figure, and meeting the opposite sides, the rectangle under their segments will be equivalent.

XCVIII. The perpendicular within a is a mean proportional between the segments formed on it by right lines, drawn from the extremities of the diameter through any point in the circumference.

XCIX. The area of a quadrilateral figure inscribed in a is a mean proportional between the rectangle of the excesses of the semi-perimeter above any two sides, and the rectangle of the excesses of the semi-perimeter above the other two sides.

C. Let lines drawn from any point p in the circumference of a to the several s of a regular inscribed polygon of an odd number of sides (n), be called in their order a1 a a3 an; then it may be proved by plane geometry, that a'+a+a3 - a" a2+a++

CI. If the polygon be of an even number of sides, then let p4, pZ, be the arc of the first and last Zs of the polygon, and pA the greater; take on pД an arc pm = 144 (pA--pZ), draw a radius to m, cutting the chord pÅ in n; then 2 Pn is the difference between a' + a3 + a3

a" and a+a+a°

END OF THEOREMS.

LOCI.

1. Given the base and vertical of a ▲, required the following loci.

1o. Of the occurse of the perpendiculars from the s on the sides.

2o. Of the centre of the circumscribing .

3°. Of the bisectors of the sides.

4°. Of the centre of the inscribed O.

5o. Of the centre of the, that touches the base and two sides produced.

6°. Of the centre of the O, that touches one side and the productions of the base and other side.

II. Given the sum of the sides, and base, to find the locus.

1o. Of the centre of the circle, that touches one side and the base, and the other side produced.

2o. Of the extremities of perpendiculars from the ends of the base, on a line through the vertical, making = Ls with the sides.

III. Given the base and difference of the sides, to find the locus..

1o. Of the centre of the inscribed O.

2o. Of the extremities of perpendiculars from the ends of the base on the line bisecting the vertical 4.

The other loci of those latter data are of too high an order for elementary geometry.

IV. Given the base and ratio of the sides, to find the locus of the vertex.

V. Given the inscribed O, the difference of the sides, and position of the base; the locus of the vertex will be in a certain right line, through the upper end of the diameter of the circle, from the point of contact of the base.

VI. If the sum of the 's of the lines from a point to two, three, four, &c. points, that are given, be of a given magnitude; required the locus of the point?

VII. If the sum of the 2s of the lines in this case be a minimum, what becomes of the locus?

VIII. Given the diameters of the inscribed and circumscribing Os, and the centre of either, to find the locus of the centre of the other.

IX. A point and a right line being given in position,

the locus of another point, the □ of whose distance from the former is to the rectangle under the distance from the latter and a given right line, is a given O.

X. If from two given points there be inflected two right lines, such that the difference of the Os of one and a given space, shall have to the of the other a given ratio; this point of concourse will lie in the circumference of a given O.

XI. If two right lines containing a given rectangle, be drawn from a point at a given ; should the one terminate in a right line given in position, the other will terminate in the circumference of a O.

XII. To find the locus of the extremity of a right line, which bisects the contained in a given segment of a O, and is to half the sum of its sides.

XIII. Let A and B be two given points in AB a right line given in position, and let C and D be two given points outside this line; let CV and DV be drawn, meeting AB in F and G, so that A F may have to B Ga given ratio; it is required to determine the locus of the point V ?

XIV. Let AB and AC be two right lines given in position; and let DE be a line of a given length terminated by them at D and E; let DV and EV be perpendiculars to the lines AB and AC meeting in V. It is required to determine the locus of the point V?

XV. A circular ring revolves on the inside of another exactly double its size; required the locus of a given point in its circumference ?

XVI. If a plane ▲ be so placed, that two of its angular points may always be on two right lines given in position; it is required to determine the locus of the remaining ?

XVII. Let ACB be a given ▲; D, L and Q, given points in the side AB; through D draw any right line DEF, cutting AC and BC in E and F respectively, and through E and F draw right lines LEP and QFP intersecting in P. Required the locus of P?

XVIII. If a right line, drawn through a given point to a right line given in position, be divided in a given ratio, the locus of the point of section is a right line given in position.

XIX. If a right line, drawn from a given point to a right line given in position, contain a given rectangle, the locus of its point of section will be a given O.

XX. If through a given point, two right lines be drawn in a given ratio, and containing a given; if the one terminate in a given circumference, the other will also terminate in a given circumference.

XXI. If a right line, drawn through a given point to the circumference of a given be divided in a given ratio, the locus of the point of section will also be the circumference of a given circle.

XXII. If two right lines, in a given ratio, stand at givens to two diverging lines, which are given in position, the locus of their vertex will likewise be a right line given in position.

XXIII. If a right line, given in position, be cut at givens by two right lines, which intercept, from two given points in it, segments, that have a given ratio, the locus of the point of concourse is a right line given in position.

XXIV. Given a point F, and a right line AB in position, drawing any right line FG to the given line, and producing it to a point D, so that GF x FD shall be a given rectangle; required the locus of D.

PORISMS.

I. Given two points in the circumference of a circle, a circle may be found, such, that, if any point of it be joined with the given points, and a tangent be drawn, the rectangle under the lines may be to the square of the tangent in a given ratio.

II. Two right lines being given in position, a circle may be found, such, that, if another circle be described upon any radius thereof as diameter, the chord of the arc of this latter, intercepted by the lines, will be of a given magnitude.

III. Let A and B be two given points outside a circle, given in magnitude and position; a point C may be found within the circle, such, that, if any right line be drawn through it, cutting the circle in F and G, and AF and BG be joined, the square of AF will have to the square of BG, a ratio which is compounded of the ratios of FC to CG, and a certain given ratio; which ratio is also to be found.