Problem 59.

To draw a tangential arc to two given circles A and B, touching one of the given circles in any given point C.

1. From C, through centre A, draw CD of unlimited length. 2. From centre B, draw BE parallel to CD (Pr. 8).

3. From C, through E, draw CF; and from F, through B, draw FG.

4. With G as centre, and GC as radius, describe the required tangential arc. Then the arc CF shall be tangential to the two given circles A and B.

Problem 60.

To find a point which would be situated in the continuation of a given arc ABC, when the centre of the arc cannot be obtained.

A

B

с

D

E

1. Draw any two chords AB, BC.

2. Make the angle BCD equal to the angle ABC (Pr. 10.)

3. Cut off CE equal to AB; then point E would be in the continuation of the given arc ABC.

Problem 61.

To describe the arc of a circle, which shall be tangential to any two given converging lines AB, CD, and which shall touch one of the given lines at a given point E.

1. Produce the converging lines AB, CD, until they meet in point F, and bisect the angle AFC by the line FG (Pr. 4).

2. From E, draw EH perpendicular to AB (Pr. 2).

3. With H as centre, and HE as radius, describe the required arc. Then the arc EK shall be tangential to the two given converging lines AB, CD.

SECTION V.-POLYGONS.

DEFINITIONS.

(In the construction of rectilineal figures, we have hitherto treated of only TRILATERAL and QUADRILATERAL figures. Sections II. and III.)

1. "Multilateral figures, or polygons, are those which are contained by more than four straight lines" (Euc. I., Def. 23).

NOTE.-A polygon is either regular or irregular.

2. A regular polygon is one that has all its sides and all its angles equal. Ex. ABCDE

E

3. An irregular polygon is one that has its sides and angles unequal. Ex. ABCDE—

B

NOTE.-A polygon may have any number of sides, but in Practical Geometry we seldom have to deal with figures having more than twelve sides.

4. A pentagon is a polygon having 5 sides.

6

A hexagon

A heptagon

An octagon

A nonagon

A decagon

An un-decagon

A do-decagon

Problem 62-(A.)

To inscribe any regular polygon (say a pentagon) in a given circle A.

General Method.

1. Draw a diameter BC, and divide it into as many equal parts as the polygon is to have sides (in this case five. Pr. 15).

B

H

2. With points B and C as centres, and the diameter BC as radius, describe arcs cutting at D.

3. From D, draw a line through point 2 to E. Join EB, which is one of the sides of the required polygon.

4. With EB as radius, starting from B, cut the circle in the points F, G, H successively.

5. Join BF, FG, GH, and HE by straight lines, and a regular pentagon will be inscribed within a given circle A.

NOTE. Whatever number of sides the polygon may have, the line from D must always be drawn through the second division of the diameter.

(B.)

To inscribe any regular polygon (say an octagon) in a given circle A.

Another General Method.

1. Draw any radius AB, and at B draw a tangent to the circle (Pr. 54).

2. From B, with any radius, describe a semicircle CDE, and divide it into as many parts as the polygon is to have sides.

6

B

3. Draw lines from B through each point of division; produce them, and they will cut the circle in the place of the angles of the polygon.

4. Join the points B 1, 12, 23, &c., and a regular octagon will be inscribed in the given circle A.

Problem 63.

To inscribe a regular pentagon within a given circle A.

1. Draw a diameter BC, and from the centre A erect a perpendicular AD (Pr. 2).

2. Bisect the radius AC in E (Pr. 1).