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2. If a polygon inscribed in a circle is equilateral, it is also equiangular.
Let AB, BC, CD be consecutive sides of an equilateral polygon inscribed in the O ADK. Then shall this polygon be equiangular.
K Because the chord AB=the chord DC. Hyp. :: the minor arc AB = the minor arc DC. III. 28. A To each of these equals add the arc AKD:
then the arc BAKD= the arc AKDC; :: the angles at the Oce, which stand on these B equal arcs, are equal ;
that is, the L BCD=the L ABC. III. 27.
Similarly the remaining angles of the polygon may be shewn to be equal : the polygon is equiangular.
3. If a polygon inscribed in a circle is equiangular, it is also equilateral, provided that the number of its sides is odd.
[Observe that Theorems 2 and 3 are only true of polygons inscribed in a circle.
The above figures are sufficient to shew that otherwise a polygon may be equilateral without being equiangular, Fig. 1; or equiangular without being equilateral, Fig. 2.]
Note. The following extensions of Euclid's constructions for Regular Polygons should be noticed.
By continual bisection of arcs, we are enabled to divide the circumference of a circle, by means of Proposition 6, into 4, 8, 16,..., 2.2",... equal parts ; by means of Proposition 15, into 3, 6, 12,. 3.2n,
equal parts; by means of Proposition 11, into 5, 10, 20,. 5. 2n
equal parts ; by means of Proposition 16, into 15, 30, 60,..., 15.2", ... equal parts.
Hence we can inscribe in a circle a regular polygon the number of whose sides is included in any one of the formulæ 2.2", 3. 2”, 5.2", 15. 2”, n being any positive integer. It has also been shewn (by Gauss, 1800) that a regular polygon of 21+1 sides may be inscribed in a circle, provided 2n + I is a prime number.
QUESTIONS FOR REVISION ON BOOK IV.
1. With what difference of meaning is the word inscribed used in the following cases ?
(i) a triangle inscribed in a circle ;
(ii) a circle inscribed in a triangle. 2. What is meant by a cyclic figure? Shew that all triangles are cyclic.
What is the condition that a quadrilateral may be cyclic ?
3. Shew that the only regular figures which may be fitted together so as to form a plane surface are (i) equilateral triangles, (ii) squares, (iii) regular hexagons. 4. Employ the first Corollary of I. 32 to shew that in any
2(n − 2) regular polygon of n sides each interior angle contains right angles ?
5. The bisectors of the angles of a regular polygon are concurrent. State the method of proof employed in this and similar theorems. 6. Shew that
(i) all squares inscribed in a given circle are equal; and
circle are equal. 7. How many circles can be described to touch each of three given straight lines of unlimited length ?
(i) when no two of the lines are parallel ;
(iii) when all three are parallel. 8. Prove that the greatest triangle which can be inscribed in a circle on a diameter as base, is one-fourth of the circumscribed square.
9. The radius of a given circle is 10 inches : find the length of a side of (i) the circumscribed square ;
[20 inches. ] (ii) the insc. ibed square ;
2 inches.] (iii) the inscribed equilateral triangle ; [1013 inches.] (iv) the circumscribed equilateral triangle ; [2013 inches.] (v) the inscribed regular hexagon.
[10 inches.] Shew also that the areas of these figures are respectively 400, 200, 751/3, 30013, and 150/3 square inches.
THEOREMS AND EXAMPLES ON BOOK IV.
ON THE TRIANGLE AND ITS CIRCLES.
1. D, E, F are the points of contact of the inscribed circle of the triangle ABC, and D, E, F the points of contact of the escribed circle, which touches BC and the other sides produced : a, b, c denote the length of the sides BC, CA, AB; s the semi-perimeter of the triangle, and r, r, the radii of the inscribed and escribed circles.
Prove the following equalities : (i) AE =AF = s-a,
BD =BF =s - b,
CD =CE =8-C,
BD,=BF1=8 - C.
=rs=r1(8 - a).
2. In the triangle ABC, I is the centre of the inscribed circle, and 11, 12, 13 the centres of the escribed circles touching respectively the sides BC, CĂ, AB and the other sides produced.
Prove the following properties :
(i) The points A, I, 1 are collinear: so are B, I, 1,; and C, 1, 1s.
(ii) The points 12, A, 13 are collinear; so are 13, B, 11; and LC,
(iii) The triangles BI,C, CI,A, Al,B are equiangular to one another.
(iv) The triangle 11elz is equiangular to the triangle formed by joining the points of contact of the inscribed circle.
(v) Of the four points 1, 11, 12, 13 each is the orthocentre of the triangle whose vertices are the other three.
(vi) The four circles, each of which passes through three of the points I, 11, 12, 13, are all equal.
3. With the notation of page 297, shew that in a triangle ABC, if the angle at C is a right angle,
r=s-c; r1=s -- b; ro=s - a; rz=s. 4. With the figure given on page 298, shew that if the circles whose centres are 1, 11, 12, 13 touch BC at D, D1, D2, D3, then (i) DD,=D,Dg=b.
(ii) DD, =D,D,=c. (iii) D.D. =b+c.
(iv) DD=bwc. 5. Shew that the orthocentre and vertices of a triangle are the centres of the inscribed and escribed circles of the pedal triangle.
[See Ex. 20, p. 243.] 6. Given the base and vertical angle of a triangle, find the locus of the centre of the inscribed circle.
[See Ex. 36, p. 246. ] 7. Given the base and vertical angle of a triangle, find the locus of the centre of the escribed circle which touches the base.
8. Given the base and vertical angle of a triangle, shew that the centre of the circumscribed circle is fixed.
9. Given the base BC, and the vertical angle A of a triangle, find the locus of the centre of thé escribed circle which touches AC.
10. Given the base, the vertical angle, and the radius of the inscribed circle ; construct the triangle.
11. Given the base, the vertical angle, and the radius of the escribed circle, (i) which touches the base, (ii) which touches one of the sides containing the vertical angle ; construct the triangle.
12. Given the base, the vertical angle, and the point of contact with the base of the inscribed circle ; construct the triangle.
13. Given the base, the vertical angle, and the point of contact with the base, or base produced, of an escribed circle ; construct the triangle.
14. From an external point A two tangents AB, AC are drawn to a given circle ; and the angle BAC is bisected by a straight line which meets the circumference in I and ly: shew that I is the centre of the circle inscribed in the triangle ABC, and I, the centre of one of the escribed circles.
15. lis the centre of the circle inscribed in a triangle, and 11, 12, 13 the centres of the escribed circles ; shew that 111, 112, 11, are bisected by the circumference of the circumscribed circle.
16. ABC is a triangle, and l.2, 13 the centres of the escribed circles which touch AC, and AB respectively: shew that the points B, C, 12, 13 lie upon a circle whose centre is on the circumference of the circle circumscribed about ABC.