In the triangles OBK, OBL,

the angle OKB is equal to the angle OLB,

the angle OBK is equal to the angle OBL, and the side OB is common to both; therefore OK is equal to OL.

In like manner OL is equal to OM, and so on.

I. Prob. 3.

Hence the perpendiculars OK, OL, OM... are all equal.

Hyp.

I. 19.

Q.E.D.

PROB. IO.

To circumscribe a circle about, or to inscribe a circle in, a given regular figure.

Let ABCD... be the given regular polygon : it is required to circumscribe a circle about, and to inscribe a circle in, the polygon ABCD

...

K

Bisect two adjacent angles A, B of the polygon, and let the bisectors meet at O,

draw OK perpendicular to AB,

describe circles with centre O, and radii OA and OK :

I. Prob. I.

I. Prob. 3.

these circles shall be the circumscribed and inscribed circles required.

Because OA, OB, OC . . . are all equal,

III. 27.

therefore B, C . . . lie on the circumference of the circle whose centre is O and radius OA; III. 1, Cor. therefore this circle is the circumscribed circle of the polygon ABCD ..

Again, all the perpendiculars from O on the sides of the polygon are equal;

III. 27.

therefore the feet of these perpendiculars lie on the circumference of the circle whose centre is O and radius OK, and each of the sides will touch this circle; therefore the circle is the inscribed circle of the polygon ABCD ...

III. 20.

Q.E.F.

PROB. II. To inscribe in, or to circumscribe about, a given circle regular figures of 4, 8, 16, 32... sides.

Let ABCD be the given circle: it is required to inscribe in, or to circumscribe about, ABCD regular figures of 4, 8, 16,

Find the centre O, and draw two diameters AC, BD, at right

angles to each other.

Join AB, BC, CD, DA,

I. Prob. 2.

and draw tangents to the circle at A, B, C, D, so as to form the figure PQRS :

III. 20, Cor. 5.

ABCD and PQRS shall be the inscribed and circumscribed regular figures of four sides.

The angles at O are all right angles;

Constr.

therefore the arcs AB, BC, CD, DA are all equal;

111. 4.

therefore the figures ABCD, PQRS are regular.

III. 26.

To inscribe, and circumscribe, regular figures of eight sides, bisect the arcs AB, BC, CD, DA,

and proceed as before.

III. Prob. 1.

By repeating this process figures of 16, 32

sides may

be drawn.

Q.E.F.

PROB. 12.

To inscribe in, or to circumscribe about, a given circle, regular figures of 3, 6, 12, 24... sides.

Let ABC be the given circle: it is required to inscribe, in, or to circumscribe about, ABC regular figures of 3, 6, 12, 24.. sides.

With any point D on the circumference of the given circle as centre, and with radius equal to the radius of the given circle, describe a circle cutting the circumference ABC at A and B ; with centre B and the same radius describe a circle cutting ABC at E, and with centre E and the same radius describe a circle cutting ABC at C.

Join AB, BC, CA,

and draw tangents at ABC so as to form the figure HKL:

ABC and HKL shall be the required inscribed and circumscribed regular figures of 3 sides.

Because each of the triangles AOD, DOB, BOE, EOC is equilateral,

Constr.

I 25.

therefore each of the angles AOD, DOB, BOE, EOC is a third of two right angles; therefore each of the angles AOB, BOC is a third of four right angles.

But all the angles round O are together equal to four right angles; therefore the remaining angle COA is also a third of four right angles;

therefore the arcs AB, BC, CA are all equal; therefore the figures ABC, HKL are regular.

III. 4.

III. 26.

Figures of 6, 12, 24 sides may be obtained by repeatedly bisecting the arcs AB, BC, CA, and drawing chords and tangents as before.

Q E.F.

Ex. 118. To describe a regular hexagon ABCDEF, such that the diagonal AC may be of given length.

EXERCISES.

119. Two circles are described, each touching one side of a triangle, and the other two sides produced. Shew that a circle can be described passing through their centres and two of the angular points of the triangle.

*120. If ABC be a triangle, shew that the circle through B, C and the centre of the escribed circle touching BC passes through the centre of the inscribed circle.

121. Through each angular point of any triangle two straight lines are drawn parallel to the lines joining the centre of the circumscribed circle to the other angles of the triangle: prove that these straight lines will form an equilateral hexagon, and that each of the angles of this hexagon is equal to one of its other angles, and double one of the angles of the triangle.

I22.

The radii of the inscribed, circumscribed, and escribed circles of an equilateral triangle are in the proportion of 1, 2, 3. 123. The equilateral triangle, square, and regular hexagon inscribed in a given circle are respectively one-fourth, one-half, and three-fourths of the corresponding figures described about it.