16. The base, the vertical angle, and the inscribed radius. 17. The perimeter, the vertical angle, and the inscribed radius. 18. The base, the sum or difference of the other two sides, and the inscribed radius. 19. Prove the following properties with respect to ▲ ABC (see fig. (6) a + b + c = AE + AE1 + AE2 + AE3 = AF + AF1 + AF2 + AF3 = BD + BD1 + BD1⁄2 + BD3 = BF + BF1 + BF2+ BF3 = CD + CD1 + CD2 + CD3 = CE CE1 + CE2 + CE3. 2 (7) a2+b2+c2 = AE2 + AE22 + AЕ„2 + AE ̧2 = = = AF2 + AF22 + AF22 + AF2 BD2 + BD12 + BD22 + BD ̧2 = BF2 + BF2 + BF22 + BF3 CD3 + CD + CD22 + CD32 = CE2 + CE12 + CE22 + CE32. AI2 + AI2 + AI2 + + CF2 + CF12 + CF22 + CF32 = · 3 (a2 + b2 + c2) + *The last four sets of expressions may be written more shortly by using the Greek letter Σ (sigma) as equivalent to 'the sum of all such terms as.' Thus (6) would be a + b + c = Σ (AE) = Σ (AF) = &c. (9) would be Σ (AD2) + Σ (BE2) + Σ (CF2) = 5 (a2 + b2 + c2). This property is due to W. H. Levy; see Lady's and Gentleman's Diary for 1852, p. 71. (10) Triangles mutually equiangular in sets of four are: AIE, AIE, AIE, AI3E3; BIF, BIF1, BI,F, BI¿Fз; (11) Mention other twelve triangles which are mutually equi- (12) Triangles mutually equiangular in sets of three are : AIB, ACI1, ICB ; BIC, BAI 2, I¿AC; CIA, CBI„, IBA. (13) Triangles mutually equiangular in sets of four are: I1Bİ, ICI, IBI,, ICI2; I¿CI3, I2AI1, ICI1, IAI;; (14) Express in terms of s A, B, C, 3, (a) The angles of ▲s I1⁄2Ï1⁄2Ï3, DEF; I¿BC, I1⁄2ÑA, I1⁄2AB; AEF, BFD, CDE. (b) (c) subtended by AR, BC, CA at I, I1, I2, I3. DE, EF, FD; I1I2, I2I3, I3I1; ID, IE, IF at I. (15) D and D1 are equidistant from the middle point of BC; so are D, and D. Similar relations hold for the E points and the F points. 20. Of the four points I, I, I, I,, any one is the orthocentre of the triangle formed by joining the other three, and in each case ABC is the orthocentric triangle. 21. The orthocentre and vertices of a triangle are the inscribed and escribed centres of its orthocentric triangle. Verify in the four cases. 22. Six straight lines join the inscribed and escribed centres; the circles described on these as diameters pass each through two vertices of the triangle, and the centres of these six circles lie on the Oce of the circle circumscribed about the triangle. 23. Prove the second part of the last deduction without assuming the property of the medioscribed circle. 24. Prove the following properties (see fig. on p. 251): (1) The radii ÃD1, I‚Æ‚ Ï¿F ̧ are concurrent at S1; ID, I1⁄2Œ3⁄4, I‚F2 at A1; I3D3, IE, I1F1 at B1; ID, I11, IF at C1. 2 (2) The figures Â ̧Ã ̧§ ̧Ã„ B ̧Ã ̧§1Ã ̧, C1Ï‚§1Ï1 are rhombi, and A1IB1ICI, is an equilateral hexagon whose opposite sides are parallel. (3) As ABC1, I12 are congruent, and their corresponding sides are parallel. (4) The points S1, A1, B1, C1, are the circumscribed centres of ▲§ 111⁄2Ï3, II,I3, II3I1, II1I2. (5) The figures А ̧ÏÂ ̧Ã ̧, В1IС11, C1IA12 are rhombi, and I is the circumscribed centre of ▲ Â ̧Ã ̧Ñ1· (6) The circumscribed circle of ▲ ABC is the medioscribed circle of As II2Iз, II2I3, II3I1, III2; AB11, S111, S111, S111; and its centre is the middle point of IS1 [See Davies' Symmetrical Properties, &c. quoted on p. 255.] 25. The area of ▲ ABC = 78 = r2 (8 - b) r3 (8 - c). 26. The bisector of the vertical angle of a triangle cuts the Oce of the circumscribed circle at a point which is equidistant from the ends of the base and from the centre of the inscribed circle. 71 (8 a) = = 27. The diameter of the circle inscribed in a right-angled triangle together with the hypotenuse = the sum of the other two sides. 28. The rectangle under the two segments of the hypotenuse of a right-angled triangle made by the point of contact of the inscribed circle: = the area of the triangle. 29. Twice the circumscribed diameter the sum of the three escribed radii diminished by the inscribed radius. 30. The sum of the distances of the circumscribed centre from the sides of a triangle = the sum of the inscribed and circumscribed radii; and the sum of the distances of the orthocentre from the vertices = the sum of the inscribed and circumscribed diameters. (Carnot's Géométrie de Position, § 137.) 31. Examine the case when the circumscribed centre and orthocentre are outside the triangle. 32. If A1B11 be the triangle formed by joining the escribed centres of ▲ ABC; A,B,C, the triangle formed by joining the escribed centres of ▲ Ã ̧В ̧Ñ1; A‚Â‚С1⁄2 the triangle formed by joining the escribed centres of ▲ A,B,C,; and this process of construction be continued, the successive triangles will approximate to an equilateral triangle. (Booth's New Geometrical Methods, vol. ii. p. 315.) 33. If an equilateral polygon be circumscribed about a circle, it will be equiangular if the number of sides be odd. Examine the case when the number of sides is even. 34. AB, CD, two alternate sides of a regular polygon, are produced to meet at E, and O is the centre of the polygon. Prove A, E, C, O concyclic, and also D, E, B, 0. 35. The sum of the perpendiculars on the sides of a regular n-gon from any point inside = n times the radius of the inscribed circle. Examine the case when the point is outside. Loci. The base and the vertical angle of a triangle are given; find the locus of 1. The orthocentre of the triangle. 2. The centre of the inscribed circle. 3. The centres of the three escribed circles. 4. The centroid of the triangle. 5. ABC is a triangle, and E is any point in AC. Through E a straight line DEF is drawn cutting AB at F and BC produced at D; circles are circumscribed about As AEF, CDE. Find the locus of the other point of intersection of the circles. 6. AB and AC are two straight lines containing a fixed angle ; and between AB and AC there is moved a straight line DE of given length. The perpendiculars from D and E to AB and AC meet at P, and the perpendiculars from D and E to AC and AB meet at 0; find the loci of O and P. 7. Given the vertical angle of a triangle, and the sum of the sides containing it; find the locus of the centre of the circle circumscribed about the triangle. 8. A circle is given, and in it are inscribed triangles, two of whose sides are respectively parallel to two fixed straight lines. Find the locus of the centres of the circles inscribed in these triangles. 9. A circle is given, and from any point P on another given con centric circle of greater radius, tangents are drawn touching the first circle at Q and R; find the loci of the centres of the inscribed and circumscribed circles of the triangle PQR. 10. A point is taken outside a square such that of the straight lines drawn from it to the vertices of the square, the two inner ones trisect the angle between the two outer ones; show that the locus of the point is the Oce of the circle circumscribed about the square. 261 BOOK V. DEFINITIONS. 1. A less magnitude is said to be a submultiple of a greater magnitude, when the less measures the greater; that is, when the less is contained a certain number of times exactly in the greater. 2. A greater magnitude is said to be a multiple of a less, when the greater is measured by the less; that is, when the greater contains the less a certain number of times exactly. 3. Equimultiples of magnitudes are multiples that contain these magnitudes, respectively, the same number of times. 4. Ratio is a relation of two magnitudes of the same kind to one another, in respect of quantuplicity (a word which refers to the number of times or parts of a time that the one is contained in the other). The two magnitudes of a ratio are called its terms. The first term is called the antecedent; the latter, the consequent. The ratio of A to B is usually expressed A: B. Of the two terms A and B, A is the antecedent, B the consequent. 5. If there be four magnitudes, such that if any equimultiples whatsoever be taken of the first and third, and any equimultiples whatsoever of the second and fourth, and if, according as the multiple of the first is greater than the multiple of the second, equal to it, or less, so is the multiple of the third greater than the multiple of the fourth, equal |