SECTION III.

OF INSCRIBING ONE FIGURE IN ANOTHER.

PROBLEM I.

To inscribe an equilateral triangle in a given circle. Let ABC be the circle in which an equilateral triangle is to be inscribed.

F

B

E

C

From any point in the circumference, as C, with the radius of the given circle, describe an arc cutting the circumference of the given circle in two points, D and E. From D to E draw the straight line DE, and it will be the side of the triangle required. If, therefore, from D as a centre with the opening DE, an arc be described cutting the given circumference in F, we have only to draw a straight line from D to F, and another from E to F, to complete the triangle.

PROBLEM II.

To inscribe a square in a given circle.

Let ABCD be a circle in which it is required to inscribe a square.

B

Draw the two diameters AD and BC, intersecting each other at right angles. Then join AB, BD, DC, and CA; the figure ABCD is the square required.

PROBLEM III.

To inscribe a circle in a given triangle.

Let ABC be the triangle in which it is required to inscribe a circle.

About the angular points B and C, with any convenient radius, describe the arcs DE and FG. Then draw the straight lines BH and CH, respectively bisecting the arcs through which they pass, and meeting in H. From H let fall a perpendicular, as HK, upon either of the sides of the triangle: and lastly, from H as a centre, with the opening HK, describe the required circle.

SECTION IV.

OF CIRCUMSCRIBING ONE FIGURE ABOUT ANOTHER.

PROBLEM I.

To circumscribe a circle about a given triangle.

Let ABC be the triangle about which it is required to circumscribe a circle.

D

Bisect any two of the sides, as AC and CB ; then from D and E, the points of bisection, raise the perpendiculars DF and EF, intersecting one another in the point F. Lastly, form F, with the opening FA, describe a circle about the given triangle.

PROBLEM II.

To circumscribe a circle about a given square.

Let ABCD be the given square.

A

B

Draw the diagonals AD, BC, intersecting each other

in E, and from E as a centre, with EA for radius, describe the circle required.

PROBLEM III.

To circumscribe a square about a given circle.

Let the circle to be circumscribed with a square ABCD.

be

Draw the diameters AB, CD, intersecting each other at right angles; and from the points A, C, B, D, as centres, with AK for radius, describe semi-circles meeting one another in the points E, F, G, and H. Lastly, draw the straight lines EF, FH, HG, and GE.

SECTION V.

OF DRAWING PROPORTIONAL LINES.

PROBLEM I.

To find a mean proportional between two given lines.

Let A and B be the lines between which a mean proportional is to be found.

Draw the indefinite straight line CD, and from C

set off CE equal to A. Then from E set off EF equal to B, and bisect CF in G. From G as a centre, with the opening GC, describe a semicircle as CHF; and lastly, raise the perpendicular EH, which will be the mean sought.

PROBLEM II.

To find a fourth proportional to three given lines.

Let the three lines to which a fourth proportional is to be found be A, B, and C.

Make at pleasure the rectilineal angle KDE, and from D towards E lay off DF equal to A. Also from D towards K lay off DG equal to B. Again, from F towards E lay off FH equal to C. Join FG, and through the point H draw HK parallel to FG. The line GK is the fourth proportional required.

PROBLEM III.

To divide a given right line in any proposed ratio.