B D 2 ..h' = _ (a’+6—(?)? APPLICATIONS. 1. To find the altitudes of a triangle in terms of its sides. * See (332) Let ABC be a A ; denote by a, b, c, the lengths of the sides opposite the _s A, B, C, respectively, and by 1 the altitude AD. Of the two Zs Band C, at least one is acute; suppose it to be the _C. Then: 4. In the A ADC, h = 73 – CD'. : (328) In the A ABC, c = a + b3 – 2a x CD. , ο a ? (330) a + 3° - C c ..CD = 2a 4*3*-(a^+j*-co) 4a? (by Alg. t) 4a (1) 4a Put a+b+c=2s. Then a + b -c= 2(s — c). s Substituting in (1) and extracting the square root, we have 2 Vs(s – a)(s – b)(s – c). 2. To find the medians of a triangle in terms of its sides. See (333). * See Traité de Géométrie, par Rouché et Comberousse, p. 139. + The difference of the squares of two numbers equals the product of the sum and difference of the two numbers. Denote by m the median on the side BC. = c bt Then, b + c = 2m + 2 = (333) . ገዝ ...m= } V2(6 + c“) – a. 3. To find the bisectors of a triangle in terms of its sides. See (343). Denote the bisector AD, figure of (343), by d, and the sides as in Exs. 1 and 2. By (343), bc = BD X DC + d'. ’ ...da = bc BD X DC. (1) BD DC BD + DC By (303), (296) b b + c ab .:. BD and DC = btc b + c Substituting these values in (1) and reducing, and demoting by 8 the semi-perimeter of the A, we have 2 Vbes (8 a). 4. To find the radius of the circumscribed circle in terms of the sides of the triangle. See (344). Denote the radius by R, figure of (344), and the sides as before. By (344), bc = 2R X AD. ас we have abc )6 5. Find the altitudes of a triangle whose sides are 13, 9, and 6. Ans. 3.641, 5.259, 7.886. 6. Find the medians of the above triangle. Ans. 4.031, 9.069, 10.770. 7. Find (1) the bisectors of the angles of the above triangle, and (2) the radius of the circumscribed circle. Ans. (1) 3.833, 7.778, 10.407; (2) 7.416. a EXERCISES, THEOREMS. 1. If two circles touch each other, either internally or externally, any two straight lines drawn from the point of contact will be cut proportionally by the circumferences. 2. If two circles intersect in P, and the tangents at P to the two circles meet the circles again in Q and R; prove that PQ: PR in the same ratio as the radii of the circles. 3. ABC, DEF are two isosceles triangles, BC, EF being the bases. If AB: BC = DE: EF, show that the triangles are similar. 4. P is a point in a diagonal of a parallelogram. EPF, GPH join points in the opposite sides of the parallelogram. EH, GF cut the diagonal in which P is. Show that EH is parallel to GF. 5. The parallel straight lines AB, CD are joined by AD, BC which intersect in 0; in AB, CD points E, F are taken such that AE : EB = DF:FC : prove that E, 0, F are collinear. (164). 6. The side BC of a triangle ABC is bisected at D, and the angles ADB, ADC are bisected by the straight lines DE, DF, meeting AB, AC at E, F respectively: show that EF is parallel to BC. Apply (303). 7. AB is a diameter of a circle, CD is a chord at right angles to it, and E is any point in CD; AE and BE are drawn and produced to cut the circle at F and G: show that the quadrilateral CFDG has any two of its sides in the same ratio as the remaining two. Apply (303). 8. In the circumference of the circle of which AB is the diameter, take any point P, and draw PC, PD on opposite sides of AP, and equally inclined to it, meeting AB at C and D: show that AC: BC = AD : BD. Apply (303) and (304). 9. From the same point A straight lines are drawn making the angles BAC, CAD, DAE each equal to half a right angle, and they are cut by a straight line BCDE, which makes BAE an isosceles triangle: show that BC or DE is a mean proportional between BE and CD. Apply (303) and (304). 10. ABCD is a quadrilateral having two of its sides AB, CD parallel; AF, CG are drawn parallel to each other meeting BC, AD, respectively, in F, G. Prove that BG is parallel to DF. 11. If D be the middle point of the side BC of a triangle ABC, and if any straight line be drawn through C meeting AD in E and AB in F; show that the ratio of AE to ED will be twice the ratio of AF to FB. 12. If O be the centre of the inscribed circle of the triangle ABC, and A0 meet BC in D; prove that AO is to OD as the sum of AB and AC is to BC. Apply (303). 13. ABCD is a quadrilateral inscribed in a circle; AD, BC meet in P, and the angle P is bisected by a straight line cutting AB, CD in E, F. Show that AB BE = CD : DF. Apply (309). 14. ABC is a straight line, and ABD, BCE triangles on the same side of it, such that the angles ABD, EBC are equal, and AB: BC = BE: BD: if AE, CD intersect in F, prove that AFC is an isosceles triangle. a : See (314). 15. Two circles touch in C, a point D is taken outside them such that the radii AC, BC subtend equal angles at D, and DE, DF are tangents to the circles: if EF cut DG in G, prove that DE : DF EG : GF. 16. C is the centre of a circle, and A any point within it; CA is produced through A to a point B such that the radius is a mean proportional between CA and CB: show that if P be any point on the circumference, the angles CPA and CPB are equal. See (314). 17. AB, AC are the equal sides of an isosceles triangle; the straight line bisecting AB at right angles meets BC in D: prove that the square on AB = BC X BD. 18. The square on the base of an isosceles triangle is equal to twice the product of either side by the part of that side intercepted between the perpendicular let fall on the side from the opposite angle and the end of the base. Apply (330). 19. ABC is a triangle having the sides AB, AC equal; if AB is produced to D so that BD is equal to AB, show that the square on CD is equal to the square on AB, together with twice the square on BC. 20. The sum of the squares on the four sides of a parallelogram is equal to the sum of the squares on the diagonals. Apply (333). 21. The base of a triangle is given and is bisected by the centre of a given circle : if the vertex be at any point of the : circumference, show that the sum of the squares on the two sides of the triangle is constant. 22. In any equilateral the sum of the squares on the diagonals is equal to the sum of the squares on the straight lines joining the middle points of opposite sides. Apply (155) and then Ex. 20. 23. The sum of the squares on the four sides of a quadrilateral is equal to the sum of the squares on the diagonals, increased by four times the square on the line joining the middle points of the diagonals. Apply (333). 24. AB is a diameter of a circle, the chords AC, BD intersect in a point E inside the circle : prove that the square on AB is equal to the sum of the products of AC, AE and BD, BE. Apply (335) and (331). 25. AD, BE, CF, the perpendiculars from the vertices on the opposite sides of a triangle ABC, intersect in 0: prove that the products of AO, OD and BO, OE and CO, OF are equal to each other. Apply (335). 26. With a point O inside a circle ABC as centre a circle is described cutting the former circle in D, E; a chord AOB is drawn to the first circle: if the product AO, OB is a |