Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

7. To find the locus of the vertex of a triangle, given one side and the length of the corresponding median.

8. To construct a triangle, given the middle points of the sides.

9. To construct a triangle, given the three medians. 10. To construct a triangle, given h, m, t.

11. To construct a triangle, given the feet of the three altitudes.

12. To inscribe a rectangle in a circle, given a+b.

13. To inscribe a rectangle in a circle, given a—b.

14. To construct three equal circles about a given circle so that each shall touch the other two, and also the given circle.

15. To construct three equal circles within a given circle so that each shall touch two others, and also the given. circle.

16. To construct four equal circles about a given circle so that each shall touch two others, and also the given circle.

17. To construct four equal circles within a given circle so that each shall touch two others, and also the given circle.

18. To construct three equal circles in an equilateral triangle so that each shall touch the other two, and also two sides of the triangle.

19. To construct four equal circles in a square so that each shall touch two others, and also two sides of the square.

20. In a given triangle to construct a semicircle having its diameter on one side, and touching the other two sides.

21. To inscribe a circle in a given sector.

22. To construct an equilateral triangle so that its vertices shall lie in three given parallel lines.

23. In a square ABCD to construct an equilateral triangle AEF, so that E and F shall lie in the sides of the square.

24. To cut off the corners of a square by straight lines in such a way that a regular octagon shall be formed.

25. Through the vertices of a given equilateral triangle to draw lines which shall form another equilateral triangle having a given side.

26. In a given square to construct a square having a given side so that its vertices shall lie in the sides of the given square.

27. To inscribe a square in the part common to two equal intersecting circles.

28. Through a given point A in the plane of a given circle any secant ABC is drawn. At the middle point M of BC a perpendicular MP equal to MA is erected. Find the locus of the point P.

29. Given a circle and two parallel secants; to draw a tangent so that the part contained between the secants. shall be bisected at the point of contact.

30. Given a circle and two lines, OA, OB, meeting at the centre O; to draw a tangent AB so that the part contained between the lines shall have a given length.

Given P, P1, P2; through P to draw a line so that, if perpendiculars PX, P, Y are dropped to the line,

31. PX+PY=a.

32. PX-PY=b.

CHAPTER III.

SIMILAR FIGURES.

§ 13. THEOREMS.

1. If three lines divide two parallels into proportional parts, these lines meet in one point.

[blocks in formation]

Let AB and CD meet in O(Fig. 45). In the similar A OAC, OBD, AC: BD

=

[ocr errors]

CO: DO.

Let CD and EF meet in Q. In the similar A QCE, QDF, CE: DF CQ: DQ. But, by the hypothesis,

[blocks in formation]

a proportion from which it follows that O and Q must coincide. For, by the Theory of Proportions,

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

2. Any two altitudes of a triangle are inversely proportional to the corresponding bases.

3. If the line joining the middle points of the bases of a trapezoid is produced, and the two legs are also produced, the three lines will meet in the same point.

4. If a line drawn from one vertex of a triangle divides the opposite side into parts proportional to the adjacent sides, the line bisects the angle at the vertex,

This theorem is the converse of No. 143. Either a direct or indirect proof may be given.

5. State and prove the converse of No. 144.

6. In a quadrilateral ABCD, in which the angles at B and D are right angles, perpendiculars PE, PF are dropped from any point P, in the diagonal AC, to the sides BC, AD, respectively. Prove that

[blocks in formation]

7. If in a triangle ABC any length AD is taken from one side AC, and an equal length BE is added to the side CB, the new base DE is divided by the base AB in the inverse ratio of the sides AC and BC.

Draw (Fig. 46) DF || to AB, and apply No. 139.

8. In a circle a line EF is drawn perpendicular to a diameter AB, and meeting it in G. Through A any chord AD is drawn, meeting EF in C. Prove that the product ADX AC is constant, whatever be the direction of AD.

Join BD (Fig. 47) and compare the ▲ ACG, ADB. Is the theorem also true when G lies outside the circle?

9. The squares of two chords, drawn through the same point in a circumference, have the same ratio as their projections upon the diameter drawn through that point. (No. 160.)

10. If two lines OA, OB, drawn through a point O, are divided in C, D, respectively, so that OAX OC=OBXOD, a circle can be described through the points A, B, C, D.

Show that (Fig. 48) the ▲ DAO, CBO are similar, and the ▲ DAO CBO equal. Therefore, if a segment be described upon CD capable of containing the Z DAO, the arc of this segment will pass through B.

[blocks in formation]

11. If in a parallelogram ABCD a secant DE is drawn, meeting the diagonal AC in F, the side BC in G, and the side AB produced in E, then DF FGX FE.

=

The ▲ AFE, DFC (Fig. 49) are similar; and also the ▲ AFD, CFG.

12. The sum of the squares of the segments formed by two perpendicular chords is equal to the square of the diameter of the circle.

(Fig. 50.) Apply No. 161; also show that EC= AD.

13. If three circles mutually intersect one another, the common chords pass through the same point.

Let M, N, R (Fig. 51) denote the circles, and let the chords CD, EF meet in O. Join AO, and suppose that AO produced does not pass through B, but through Pin M and Q in N. Then we have,

« ΠροηγούμενηΣυνέχεια »