A with the distance BF cut the circle in G and H, inflect the chords GH and GI equal to the radius AB, and from the points H and I, with distance BF or AG, describe arcs intersecting in L.

For BL is the greater segment of the radius BH divided by a medial section; wherefore AL is equal to the side of the inscribed pentagon, and BL, to that of the decagon inscribed in the given circle. Hence AL may be inflected five times in the circumference, and BL ten times;

and consequently the are MK, or the excess of the fourth above the fifth, is equal to the twentieth part of the whole circumference.

PROBLEM XLIII.

To describe a regular pentagon, hexagon, or octagon, about a circle.

In the given circle inscribe a regular polygon of the same name or number of sides, as ABCDE, by one of the foregoing problems. Then to all its angular points draw tangents (by prob. 22), and these will form the circumscribing polygon required.

K

Ε

A

H

B

For all the chords, or sides of the inscribed figure, AB, BC, &c., being equal; and all the radii OA, OB, &c., being equal; all the vertical angles about the point O are equal. But the angles OEF, OAF, OAG, OBG, made by the tangents and radii, are right angles; therefore OEF + OAF = two right angles, and OAG + OBG = two right angles; consequently, also, AOE + AFE = and AOB + AGB = two right angles (cor. 2, th. 18). angles AOE+ AFE being = AOB + AGB, of which consequently, the remaining angles F and G are also equal. manner it is shown, that all the angles F, G, H, I, K, are equal.

two right angles, Hence, then, the AOB is = AOE; In the same

Again, the tangents from the same point FE, FA, are equal, as also the tangents AG, GB (cor. 2, th. 61); and the angles F and G of the isosceles triangles AFE, AGB, are equal, as well as their opposite sides AE, AB; consequently, those two triangles are identical (th. 1), and have their other sides EF, FA, AG, GB, all equal, and FG equal to the double of any one of them. In like manner it is shown, that all the other sides GH, HI, IK, KF, are equal to FG, or double of the tangents GB, BH, &c.

Hence, then, the circumscribed figure is both equilateral and equiangular; which was to be shown.

Cor. The inscribed circle touches the middle of the sides of the polygon.

PROBLEM XLIV.

To inscribe a circle in a regular polygon.

Bisect any two sides of the polygon by the perpendiculars GO, FO, and their intersection O will be the centre of the inscribed circle, and OG or OF will be the radius.

For the perpendiculars to the tangents AF, AG, pass through the centre (cor., th. 47); and the inscribed circle touches the middle point F, G, by the last corollary. Also, the two sides AG, AO, of the right-angled triangle AOG, being equal to the two sides AF, AO, of the right-angled triangle AOF, the third sides OF, OG, will also be equal (cor., th. 45). Therefore, the circle described with the centre O and radius OG will pass through F, and will touch the sides in the points G and F. And the same for all the other sides of the figure.

PROBLEM XLV.

To describe a circle about a regular polygon.

Bisect any two of the angles C and D with the lines CO, DO; then their intersection O will be the centre of the circumscribing circle; and OC, or OD, will be the radius.

For, draw OB, OA, OE, &c., to the angular points of the given polygon. Then the triangle OCD is isosceles, having the angles at C and D equal, being the halves of the equal angles of the polygon BCD, CDE; therefore, their opposite sides CO, DO, are equal (th. 4). But the two triangles OCD, OCB, having the two sides OC, CD, equal to the two OC, CB, and the included angles OCD, OCB, also equal, will be identical (th. 1), and have their third sides BO, OD, equal. In like manner it is shown, that all the lines OA, OB, OC, OD, OE, are equal. Consequently, a circle described with the centre O and radius OA, will pass through all the other angular points, B, C, D, &c., and will circumscribe the polygon.

PROBLEM XLVI.

On a given line to construct a rectilinear figure similar to a given rectilinear figure.

Let abcde be the given rectilinear figure, and AB the side of the proposed similar figure that is similarly posited with ab.

Place AB in the prolongation of ab, or parallel to it. Draw AC, AD, AE, &c., parallel to ac, ad, ae, respectively. Draw BC parallel to bc, meeting AC in C; CD

E

a

parallel to cd, and meeting AD in D; DE parallel to de, and meeting AE in E; and so on, till the figure is completed. Then ABCDE will be similar to abcde, from the nature of parallel lines and similar figures (th. 89).

MISCELLANEOUS EXERCISES IN PLANE GEOMETRY.

(1.) From two given points, to draw two equal straight lines, which shall meet in the same point of a line given in position.

(2.) From two given points, on the same side, or opposite sides of a line given in position, to draw two lines, which shall meet in that line, and make equal angles with it.

(3.) To trisect a given finite straight line.

(4.) If from the extremities of the diameter of a semicircle, perpendiculars be let fall on any line cutting the semicircle, the parts intercepted between those perpendiculars and the circumference are equal.

(5.) If on each side of any point in a circle any number of equal arcs be taken, and the extremities of each pair joined, the sum of the chords so drawn will be equal to the last chord produced to meet a line drawn from the given point through the extremity of the first arc.

(6.) If one circle touch another externally or internally, any straight line drawn through the point of contact will cut off similar segments.

(7.) If two circles touch each other, and also touch a straight line, the part of the line between the points of contact is a mean proportional between the diameters of the circles.

(8.) From two given points in the circumference of a given circle, to draw two lines to a point in the circumference, which shall cut a line given in position, so that the part of it intercepted by them may be equal to a given line.

(9.) If from any point within an equilateral triangle perpendiculars be drawn to the sides, they are, together, equal to a perpendicular drawn from any of the angles to the opposite side.

(10.) If the three sides of a triangle be bisected, the perpendiculars drawn to the sides, at the three points of bisection, will meet in the same point.

(11.) If from the three angles of a triangle lines be drawn to the points of bisection of the opposite sides, these lines intersect each other in the same point.

(12.) The three straight lines which bisect the three angles of a triangle, meet in the same point.

(13.) If from the angles of a triangle perpendiculars be drawn to the opposite sides, they will intersect in the same point.

(14.) If any two chords be drawn in a circle, to intersect at right angles, the sum of the squares of the four segments is equal to the square of the diameter of the circle.

(15.) In a given triangle to inscribe the greatest square.

(16.) In a given triangle to inscribe a rectangle, whose sides shall have a given ratio.

D D

(17.) The two sides of a triangle are, together, greater than the double of the straight line which joins the vertex and the bisection of the base.

(18.) If in the sides of a square, at equal distances from the four angles, four other points be taken, one in each side, the figure contained by the straight lines which join them shall also be a square.

(19.) If the sides of an equilateral and equiangular pentagon be produced to meet, the angles formed by these lines are, together, equal to two right angles.

(20.) If the sides of an equilateral and equiangular hexagon be produced to meet, the angles formed by these lines are, together, equal to four right angles.

(21.) If squares be described on the three sides of a right-angled triangle, and the extremities of the adjacent sides be joined, the triangles so formed are equal to the given triangle, and to each other.

(22.) If squares be described on the hypothenuse and sides of a rightangled triangle, and the extremities of the sides of the former, and the adjacent sides of the others, be joined, the sum of the squares of the lines joining them will be equal to five times the square of the hypothenuse.

(23.) To bisect a triangle by a line drawn parallel to one of its sides. (24.) To divide a circle into any number of concentric equal annuli.

(25.) To inscribe a square in a given semicircle.

(26.) If in a right-angled triangle a perpendicular be drawn from the right angle to the hypothenuse, and circles inscribed in the triangles on each side of it, their diameters will be to each other as the subtending sides of the rightangled triangle.

(27.) If on one side of an equilateral triangle, as a diameter, a semicircle be described, and from the opposite angle two straight lines be drawn to trisect that side, these lines produced will trisect the semi-circumference.

(28.) Draw straight lines across the angles of a given square, so as to form an equilateral and equiangular octagon.

(29.) The square of the side of an equilateral triangle, inscribed in a circle, is equal to three times the square of the radius.

(30.) To draw straight lines from the extremities of a chord to a point in the circumference of the circle, so that their sum shall be equal to a given line.

(31.) In a given triangle to inscribe a rectangle of a given magnitude.

(32.) Given the perimeter of a right-angled triangle, and the perpendicular from the right angle upon the hypothenuse, to construct the triangle.

(33.) Describe a circle touching a given straight line, and also passing through two given points.

(34.) In an isosceles triangle to inscribe three circles, touching each other, and each touching two of the three sides of the triangle.

GEOMETRY OF PLANES.

DEFINITIONS.

1. A PLANE is a surface in which, if any two points be taken, the straight line which joins these points will be wholly in that surface.

2. A straight line is said to be perpendicular to a plane, when it is perpendicular to all the straight lines in the plane which pass through the point in which it meets the plane.

This point is called the foot of the perpendicular.

3. The inclination of a straight line to a plane, is the acute angle contained by the straight line, and another straight line drawn from the point in which the first meets the plane, to the point in which a perpendicular to the plane, drawn from any point in the first line, meets the plane.

4. A straight line is said to be parallel to a plane when it cannot meet the plane, to whatever distance both be produced.

5. It will be proved in Prop. 2, that the common intersection of two planes is a straight line; this being premised,

The angle contained by two planes, which cut one another, is measured by the angle contained by two straight lines drawn from any point in the common intersection of the planes perpendicular to it, one in each of the planes.

This angle may be acute, right, or obtuse.

If it be a right angle, the planes are said to be perpendicular to each other. 6. Two planes are parallel to each other, when they cannot meet, to whatever distance both be produced.

PROP. I.

A straight line cannot be partly in a plane, and partly out of it.

For, by def. (1), when a straight line has two points common to a plane, it lies wholly in that plane.

PROP. II.

If two planes cut each other, their common intersection is a straight line.

Let the two planes, AB, CD, cut one another, and

let P, Q, be two points in their common section. Join P, Q;

Then, since the points P, Q, are in the same plane AB, the straight line PQ which joins them must lie wholly in that plane.

For a similar reason, PQ must lie wholly in the plane CD.

.. The straight line PQ is common to the two planes, and is .. their common intersection.