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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.
tangential arc. Then the arc CF shall be tangential to the two given circles A and B.
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.
1. Draw any two chords AB, BC.
continuation of the given arc ABC.
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 AFĆ 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.
(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
3. An irregular polygon is one that has its sides and angles
unequal. Ex. ABCDE
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.
7 An octagon
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).
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.
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.
3. Draw lines from B through each point of division; pro
duce them, and they will cut the circle in the place of
the angles of the polygon. 4. Join the points B 1, 1 2, 23, &c., and a regular octagon
will be inscribed in the given circle A.
To inscribe a regular pentagon within a given circle A.
1. Draw a diameter BC, and from the centre A erect a per
pendicular AD (Pr. 2).