LXXXVI. Let APBC be a parallelogram. Draw AB one of the diagonals, and from P, draw any line cutting the diagonal in E, and the other two sides of the parallelogram in D and d. Then ED × Ed = PE.2

LXXXVII. If two Os have the same centre, and an equilateral ▲ be described about either of them, and from any point in the circumference of the other O, perpendiculars be drawn to the sides of the A; the straight line which joins the bottoms of the perpendiculars shall form a A given in magnitude.

LXXXVIII. Suppose a quadrilateral figure be inscribed in another quadrilateral figure, so that the perime-ter 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 rrr" be the radii of the three Os that touch, each one side of a ▲ and the other two produced; then rxr+rxr" +rlxr"= s2 where s is the semipe

rimeter.

=

XC. If a be inscribed in a 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 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 directum.

XCII. Let AB be the side of a hexagon, inscribed in the DAB, and let ABC be an equilateral A, described on AB, so that the C is without the O; let AD,DB be right lines inflected to any point in the circumference, and let CE be drawn parallel to AD meeting BD in E; the D's of AD and CE are together to the 2 of BD and and twice the O2 of AB, the radius of the O.

XCIII. If from any number of given points A, A', A,” &c. situated in a right line, without 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, straight 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 A to cut the opposite sides proportionally, another straight 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 straight lines be drawn from any point in the circumference, to meet the opposite side produced both ways, the altitude of the rectangle 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 O is a mean proportional to the segments formed on it by straight 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 semiperimeter above any two sides, and the rectangle of the excesses of the semiperimeter 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 a, a, a z an; then it may be proved by plain geometry, that a, +a,+a,

2

ana2+a+

2

CI. If the polygon be of an even number of sides, then let pA, p2, be the arc of the first and lasts of the polygon, and pA the greater; take on pA an arc pm = (p×p2), draw a radius to m cutting the chord pA in n, then 2 Pn is the difference between a, ta1ta, an- and a2+aita.

ап

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 O.

3°. Of the bisectors of the sides.

4°. Of the centre of the inscribed

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

6o. Of the centre of the 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 <s with the sides.

=

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

1o. Of the centre of the inscribed O.

20. Of the extremities of perpendiculars from the ends of the base on the line bisecting the vertical.

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 O 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 's 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 straight 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 straight line — is a given O.

X. If from two given points there be inflected two straight lines, such that the difference of the 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 straight line given in position, the other will terminate in the circumference of a O.

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

sides.

XIII. Let A, B, be two given points in AB a right line given in position, and let C, D be two given points without this line; let CV, DV be drawn, meeting AB in F and G so that A F may have to BG a given ratio; it is required to determine the locus of the point V?

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

XVII. Let ACB be a given a ; D, L, Q, given points in the side AB; through D draw any right line DEF, cutting AC, BC in E and F respectively, and through

I

E and F draw right lines LEP, 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 straight 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 given s 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, seginents 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 GFX 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 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