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PROPOSITION XIV. PROBLEM.
To describe a circle about a given regular pentagon.
Let ABCDE be the given regular pentagon.
It is required to describe a ✪ about the pentagon.
Bisect the s BCD, CDE by the st. lines CO, DO, meeting in 0.
Join OB, OA, OE.
Then it may be shewn, as in the preceding Proposition, that
and OCD half ▲ BCD, and ▲ ODC=half ▲ CDE,
In the same way we may shew that OB, OA, OE
.. OA, OB, OC, OD, OE are all equal,
described with centre O and radius OA will pass
through B, C, D, E,
and will be described about the pentagon.
Q. E. F.
To inscribe a regular hexagon in a given circle.
Let ABCDEF be the given O, of which O is the centre. It is required to inscribe a regular hexagon in the ○. Draw the diameter AOD,
and with centre D and radius DO describe a O EOCG Join EO, CO, and produce them to B and F.
Join AB, BC, CD, DE, EF, FA.
ACE, .. OE=OD;
O is the centre of
D is the centre of
GCE, .. OD=DE ;
.. OED is an equilateral ▲,
EOD=the third part of two rt. 4 s. So also 4 DOC=the third part of two rt. 4 s, and.. BOC=the third part of two rt. 4 s. Thus 48 EOD, DOC, BOC are all equal; and to these the vertically opposite s BOA, AOF, FOE are equal;
.. ¿s AOB, BOC, COD, DOE, EOF, FOA, are all equal, and .. arcs AB, BC, CD, DE, EF, FA are all equal.
and.. chords AB, BC, CD, DE, EF, FA are all equal.
Thus the hexagon ABCDEF is equilateral. Also each of its 4 s-two-thirds of two rt. 4s,
.. the hexagon ABCDEF is equiangular.
Thus a regular hexagon has been inscribed in the ©.
Q. E. F.
PROPOSITION XVI. PROBLEM.
To inscribe a regular quindecagon in a given circle.
Let ABC be the given .
It is required to inscribe in the a regular quindecagon. Let AB be the side of an equilateral ▲ inscribed in the O,
and AD the side of a regular pentagon inscribed in the .
Then of such equal parts as the whole Oce ABC contains fifteen,
arc ADB must contain five,
and arc AD must contain three,
and .. arc DB, their difference, must contain two.
Bisect arc DB in E.
Then arcs DE, EB are each the fifteenth part of the whole
If then chords DE, EB be drawn,
and chords equal to them be placed all round the Oce, IV. 1. a regular quindecagon will be inscribed in the O.
Q. E. F.
Miscellaneous Exercises on Book IV.
1. The perpendiculars let fall on the sides of an equilateral triangle from the centre of the circle, described about the triangle, are equal.
2. Inscribe a circle in a given regular octagon.
3. Shew that in the diagram of Prop. X. there is a second triangle, which has each of two of its angles double of the third. 4. Describe a circle about a given rectangle.
5. Shew that the diameter of the circle which is described about an isosceles triangle, which has its vertical angle double of either of the angles at the base, is equal to the base of the triangle.
6. The side of the equilateral triangle, described about a circle, is double of the side of the equilateral triangle, inscribed in the circle.
7. A quadrilateral figure may have a circle described about it, if the rectangles contained by the segments of the diagonals be equal.
8. The square on the side of an equilateral triangle, inscribed in a circle, is triple of the square on the side of the regular hexagon, inscribed in the same circle.
9. Inscribe a circle in a given rhombus.
10. ABC is an equilateral triangle inscribed in a circle; tangents to the circle at A and B meet in M. Shew that a diameter drawn from M bisects the angle AMB, and is itself trisected by the circumference.
11. Compare the areas of two regular hexagons, one inscribed in, the other described about, a given circle.
12. Inscribe a square in a given semicircle.
13. A circle being given, describe six other circles, each of them equal to it, and in contact with each other and with the viven circle.
14. Given the angles of a triangle, and the perpendiculars from any point on the three sides, construct the triangle.
15. Having given the radius of a circle, determine its centre, when the circle touches two given lines, which are not parallel.
16. If the distance between the centres of two circles, which cut one another at right angles, is equal to twice one of the radii, the common chord is the side of the regular hexagon, inscribed in one of the circles, and the side of the equilateral triangle, inscribed in the other.
17. Construct a square, having given the sum, or the difference, of the diagonal and the side.
18. If from O, the centre of the circle inscribed in a triangle ABC, OD, OE, OF be drawn perpendicular to the sides BC, CA, AB, respectively, and from any point P in OP, drawn parallel to AB, perpendiculars PQ, PR be drawn upon OD and OE respectively, or these produced, shew that the triangle QRO is equiangular to the triangle ABC.