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Therefore BA: AC= BD:DC.
Conversely, if BA : AC=BD:DC, DA bisects FAC.

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Cor. If AD, AE bisect the interior and exterior angles, at A,

BA:AC= BD:DC, also

BA: AC= BE: EC.
Therefore BD:DC = BE: EC,

BE: EC= BE- ED: ED - EC.
That is EB, ED, EC are in Harmonic Progression.
Similarly BE, BC, BD are in Harmonic Progression.

or

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 B C produced such that BE: EC=BD: DC.

н

B

Draw any line BF. Join FD and produce it, through C draw GCH parallel to BF, meeting FD in G.

Make CH = (G. Join FH. Then since BF is not equal to CI, BC, FH being produced must meet in some point E. Then

BE:EC=FB:HC

= FB: GC

=BD:DC.
The line BC'is said to be harmonically divided.

[IV. 2.

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THEOREX III.

ICU le the midde point of a line AB which is divided harmonically in cand n, then shail the square on AJI be cual to the rectangie V'. 1).

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THEOREMS. 1. AB, AC, AD, are lines drawn through A, EFG, KHL are parallel lines meeting them. Shew that

EF: FG=KH : HL, 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 a 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, ABC" are triangles having equal angles at B, and at the angles C and c' supplementary. Then

BA: AC= BA': A'C".

Α.

A

B

G 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
BA: AG=BA: A'C'.

[1v. 2. Therefore BA: AC = BA: A'C'.

7. If two triangles hare one angle equal, and the sides about a second angle proportional, their remaining angles are either equal or supplementary.

Vake the same ounstruction as in & Shew that AC=AG. Then C either cincides with 4G, or is equally remote from the perpendiealar on the che sile. Compare I Amhinus case, and Constructions Li, 2.

Hence ACB. ICB be either both acate or both obtase, or if one of them be a riget ægle, the triangles are

à 4Bc is a triangle a ine is drawn meeting BC, CA, 4R o those pointed in D. E ad F. Sber that

47.BCE=FB.IC.A. BC is an esateral triage. Pa point lying between Band on the conjecence of the circumscribed

(II. 10, IT. 3. 13. In Construction & show that the triangles ABD, 4077, 012297 Pomariana: also that

479:0327=80. On 43:56. on AC. 71. The gemistelt des bà mon the perpendicular of an. ecpilatora siangle is to chat on the side as 3 to 4

12 the centre of a Circle its radins. On AO

diameter annihe circle i desaihed, and any common chard diant dinh A. Shew that the segments which is cuts of from the wr circles are in the ratio of 4 10 1.

1: mer hot named that segments Fhich contain equal angles are similar tigues

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 : BA: AC as BC: BD: DC.

18. ABC is any triangle, AD the perpendicular on BC; shew that

AB - ACP = DB - DO?.

19. Find the area of the triangle whose sides are 17, 15, and 8.

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