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1. Show that an equilateral polygon circumscribed about a circle is regular if the number of its sides be odd.
2. Show that an equiangular polygon inscribed in a circle is regular if the number of its sides be odd.
3. Show that any equiangular polygon circumscribed about a circle is regular.
4. Show that the side of a circumscribed equilateral triangle is double the side of an inscribed equilateral triangle.
5. Show that the area of a regular inscribed hexagon is three-fourths of that of the regular circumscribed hexagon.
6. Show that the area of a regular inscribed hexagon is a mean proportional between the areas of the inscribed and circumscribed equilateral triangles.
7. Show that the area of a regular inscribed octagon is equal to that of a rectangle whose adjacent sides are equal to the sides of the inscribed and circumscribed squares.
8. Show that the area of a regular inscribed dodecagon is equal to three times the square on the radius.
9. Given the diameter of a circle 50; find the area of the circle. Also, find the area of a sector of 80° of this circle.
10. Three equal circles touch each other externally and thus inclose one acre of ground; find the radius in rods of each of these circles.
11. Show that in two circles of different radii, angles at the centres subtended by arcs of equal length are to each other inversely as the radii.
12. Show that the square on the side of a regular inscribed pentagon, minus the square on the side of a regular inscribed decagon, is equal to the square on the radius.
PROPOSITION XIII. PROBLEM.
387. To inscribe a regular polygon of any number of sides in a given circle.
Let Q be the given circle, and n the number of sides
of the polygon.
It is required to inscribe in Q, a regular polygon having n sides.
Divide the circumference of the O iuto n equal arcs.
Join the extremities of these arcs.
Then we have the polygon required.
For the polygon is equilateral,
and the polygon is also regular,
Q. E. F.
PROPOSITION XIV. PROBLEM.
388. To inscribe in a given circle a regular polygon which has double the number of sides of a given inscribed regular polygon.
Let A B C D be the given inscribed polygon.
It is required to inscribe a regular polygon having double the number of sides of A B C D. Bisect the arcs A B, B C, etc.
Draw A E, EB, B F, etc.,
(being sides of a regular polygon).
Hence the halves of these arcs are equal,
§ 364 (an equilateral polygon inscribed in a О is regular); and has double the number of sides of the given regular polygon.
Q. E. F.
PROPOSITION XV. PROBLEM. 389. To inscribe a square in a given circle.
Let o be the centre of the given circle.
It is required to inscribe a square in the circle.
Draw the two diameters A C and BD I to each other.
Join A B, BC, C D, and DA.
Then A B C D is the square required.
For, the L A B C, BC D, etc., are rt. 4,
(being inscribed in a semicircle),
.. the figure A B C D is a square,
Q. E. F.
390. COROLLARY. By bisecting the arcs A B, BC, etc., a regular polygon of 8 sides may be inscribed ; and, by continuing the process, regular polygons of 16, 32, 64, etc., sides may be inscribed.
PROPOSITION XVI. PROBLEM. . 391. To inscribe in a given circle a regular hexagon.
Let O be the centre of the given circle.
From C as a centre, with a radius equal to 0 C,
Draw 0 F and C F.
Cons. and equiangular,
§ 112 .. the Z FOC is } of 2 rt. 4, or, $ of 4 rt. . $ 98
.. the arc FC is į of the circumference A BCF,
.. the chord FC, which subtends the arc FC, is a side of a regular hexagon ;
and the figure CFD, etc., formed by applying the radius six times as a chord, is the hexagon required.
Q. E. F. 392. COROLLARY 1. By joining the alternate vertices A, C, D, an equilateral A is inscribed in a circle.
393. Cor. 2. By bisecting the arcs A B, B C, etc., a regular polygon of 12 sides may be inscribed in a circle ; and, by continuing the process, regular polygons of 24, 48, etc., sides may be inscribed.