PLANIMETRY. ll 
(pl. 2, fig. 84). 2. To bisect a circular arc. 3. To araw a tangent to a 
given point in a circumference (pl. 2, fig. 85). 4. From a given point out 
of a circle, to draw a tangent to the circle (pil. 3, fig. 80). 5. Upon a given 
base, ab, to construct a triangle in which the angle opposite the base is equal 
to the given angle m (fig. 76). This problem is indefinite, since every 
point of an arc may be taken as the vertex of the triangle ; but it becomes 
definite if the height of the triangle is also given. A particular case is 
exhilited in the problem: upon a given line, as hypothenuse, to construct 
a right angled triangle. 6. To construct a mean proportional to two 
given lines. 7. To divide a given triangle, abc, by lines running parallel 
to a given side, into a certain number of of parts, five for instance, that shall 
be either equal, or in a definite proportion (jig. 79). 
The construction of regular polygons in and about the circle, is of impor- 
tance in understanding its theory. A regular polygon is said to be inscribed 
in the circle, when all its sides are chords; and circumscribed about the 
circle, when all its sides are tangents. A regular polygon is inscribed in a 
circle, by dividing the circumference of the latter into as many equal parts 
as the polygon is to have sides, and connecting these points by chords. 
The difficulty here lies only in dividing the circumference into a given 
number of equal parts. The division into four or six parts is most easily 
made; the former, by drawing two diameters perpendicular to each other ; 
the latter, by using as chords, lines equal to the radius (fig. 81). To divide 
the circumference into ten equal parts, we draw two radii perpendicular to 
each other, bisect the one, and connect the point of bisection with the 
extremity of the other, and then cut off from this connecting line a section 
equal to the half of the radius; the remainder will be the length of a chord 
whose arc is the tenth part of the circumference, or the side of a regular 
inscribed decagon. Pi. 2, fig. 81, shows the construction of a regular 
pentagon in a circle. AB is here a diameter, CD a radius perpendicular 
toit; from # the middle point of BC, with a radius equal to FD, we describe 
an arc intersecting AC in G; draw DG; this will be the side of the regu- 
lar pentagon (CG will be the side of the regular decagon). We may obtain 
a pentagon by connecting the alternate angles of a decagon. From the divi- 
sion into four equal parts, we may readily obtain that into 8, 16, 32, &c., and 
the division into 10, 20,40, 80, &c. The fifteenth part of the circumference 
is found by subtracting the 10th part from the 6th, for !—-3, = 4, = i. 
A regular polygon is circumscribed about a tele by dividing the cir- 
cumference into as many equal parts as the polygon is to have sides, and 
then drawing tangents to all the points of division. From a polygon of any 
given number of sides inscribed in the circle, we may obtain a regular 
polygon of double the number of sides, by bisecting the arcs, whose chords 
form the sides of the former, and drawing chords to the half-ares. The 
circumference (as well as the area) of a circle is always greater than the 
perimeter (or area) of an inscribed polygon, but is less than the perimeter 
(or area) of one circumscribed about it (pi. 3, fig. 83). 
The circumference of a circle cannot be directly measured, since it is 
not a straight line; but if two polygons of a great number of sides be 
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