PROPOSITIONS OF GEOMETRY Table 2d. Geometric Constructions A Machinery's Handbook, 31st Edition
69
B
To circumscribe a circle about a square and to inscribe a circle in a square ABCD : Draw the square’s diagonals AC and BD with the straight edge. The centers of both the circumscribed and inscribed circles are located at the point E , where the two diagonals of the square intersect. The radius of the circumscribed circle is AE , and of the inscribed circle, EF .
F
E
D
C
D
E
To inscribe a hexagon in a circle with center C : Draw diameter AB . With A and B as centers and with the circle’s radius as radius, describe circular arcs intersecting the given circle at D , E , F , and G . Draw chords AD , DE , etc., forming the required hexagon ABCDEFG .
A
B
C
F
G
To circumscribe a hexagon about a circle with center C : Draw the circle’s diameter AB , and with A as center and the radius of the circle as radius, cut the circumference of the given circle at D . Draw chord AD and bisect it with radius CE . Through E , draw FG parallel to AD and intersecting diameter AB at F . With C as center and CF as radius, draw a circle. Within this circle, inscribe the hexagon as in the preceding problem. This is the circumscribed hexagon about the first circle. To describe an ellipse with the given axes AB and CD : Describe circles with O as center and AB and CD as diameters. From a number of points, E , F , G , etc., on the outer circle, draw radii intersecting the inner circle at e , f , and g . From E , F , and G , draw lines perpendicular to AB , and from e , f , and g , draw line segments parallel to AB . The intersections of these perpendicular and parallel lines are points on the curve of the ellipse. To construct an approximate ellipse by circular arcs: Let AC be the major axis and BN the minor. With the compass, draw semicircle ADC with O as center. Divide BD into three equal parts and set off BE equal to one of these parts. With A and C as centers and OE as radius, describe circular arcs KLM and FGH ; with G and L as centers, and with the same radius, describe arcs FCH and KAM . Through F and G , draw line FP , and with P as center draw arc FBK . Arc HNM is drawn in the same manner.
A C B
F
E
D
G
E
F
D
G
e
f
g
A
B
O
C
D
L G O N E B
F
K
A
C
M
H
P
Copyright 2020, Industrial Press, Inc.
ebooks.industrialpress.com
Made with FlippingBook - Share PDF online