Conversely, if BA: AC-BD: DC, DA bisects FAC. COR. If AD, AE bisect the interior and exterior angles, at A, also or Therefore BD: DC= BE: EC, BE: EC: = BE-ED: ED – EC. That is EB, ED, EC are in Harmonic Progression. Harmonic Section. THEOREM XXI. If a given straight line BC be divided in any ratio not. equal to unity in the point D, another point E may be found in BC produced such that BE: EC = BD: DC. Draw any line BF. Join FD and produce it, through C draw GCH parallel to BF, meeting FD in G. Make CH = CG. Join FH. Then since BF is not equal to CH, BC, FH being produced must meet in some point E. BE: EC-FB: HC Then = FB: GC =BD: DC. [Iv. 2. [IV. 2. The line BC is said to be harmonically divided. THEOREM XXII. If M be the middle point of a line AB which is divided harmonically in C and D, then shall the square on AM be equal to the rectangle MC, MD. THEOREMS. 1. AB, AC, AD, are lines drawn through A, EFG, KHL are parallel lines meeting them. Shew that and find all the proportions which the lines of the figure afford. 2. Any three lines are cut by three parallel lines, shew that they are divided proportionally. 3. AB, AC are drawn through A; from B draw BC to any point C on AC, and from C draw CD to any point D on AB. Draw DE parallel to BC, and EF parallel to CD. Shew that AD is a mean proportional between AB and AF. 4. The distance of à point P from a given line AB is always in a constant ratio to its distance from another line AC; find the locus of P. 5. From points on the side of an equilateral triangle at distances 2, 4, 8 from one of the base angles perpendiculars are let fall on the base. Find the lengths which they intercept. 6. ABC, A'BC' are triangles having equal angles at B, and at the angles C and C' supplementary. Then Place the triangles so that their equal angles coincide, and through A draw AG parallel to A'C'. Then AGC=A'C'C=ACG, and AG = AC. But Therefore BA: AG=BA' : A'C'. BA: AC=BA' : A'C'. [IV. 2. 7. If two triangles have one angle equal, and the sides about a second angle proportional, their remaining angles are either equal or supplementary. Make the same construction as in 6. Shew that AC=AG. Then 40 either coincides with AG, or is equally remote from the perpendicular on the other side. Compare I. Ambignous case, and Constructions I 7, 2. Hence if 4CB, A CB be either both acute or both obtase, or if one of them be a right angle, the triangles are similar. S. 430 is a triangle, a line is drawn meeting BC, CA, 43. or those produced in D. E, and F. Shew that AF. ED. CE=FB.DC. EA. 9. 450 is an equilateral triangle. Pa point lying beTween B and C on the circumference of the circumscribed Show that P4=FB + PC. [11. 10, IV. 3. 10. In Construction &. shew that the triangles ABD, 407, CBD et proportional; also that 43D: CBD=80, or 43: 80, on 40. 11. The semicircle described upon the perpendicular of an equilateral triangle is to that on the side as 3 to 4. 12. is the centre of a circle. 40 its radius. On 40 as diameter another circle is described, and any common chord is drawn through 4. Show that the segments which it eus off from the twe eireles are in the ratio of 4 to 1. I may he assumed that segments which contain equal angles are similar figures. 13. AB is the diameter of a circle, CD a perpendicular upon it from any point in the circumference. The semicircle. on AB is equal to the semicircles on AC and CB together with the circle on CD. 14. ABC is a triangle, D, E, F, the middle points of BC, CA, AB. 1. If AD = BE then A = B. 2. If GH be any line parallel to BC the locus of its middle point is AD. 3. The locus of the intersection of BH and GC is also AD. 4. The triangle constructed with the sides AD, BE, CF is to the original triangle as 3 to 4. 5. The sum of the squares on AD, BE, CF is to the sum of the squares on the sides of the triangle as · 3 to 4. 15. A triangle may be divided into three equal parts by lines drawn from a point within it to the angles. 16. ABC is an equilateral triangle, AD the perpendicular from A, DG the perpendicular from D on AB. Shew that GB is one-fourth of AB. Determine the ratio of the squares on AG and AD. 17. ABC is a triangle, A a right angle, AD the perpendicular from A on BC. Shew that BC2: BA2: AC2 as BC: BD: DC. 18. ABC is any triangle, AD the perpendicular on BC; shew that AB-AC2= DB2 - DC2. 19. Find the area of the triangle whose sides are 17, 15, and 8. |