Construction As in the first case, except that XY is measured in the

opposite direction to AX.

Note that fig. 132 applies to the case in which p>q and fig. 133 to the case in which

q> p.

To find the fourth proportional to three given straight lines.

Let a, b, c be the three given straight lines.

Construction Make an angle POQ. From OP cut off OD = a, and

OE b. From OQ cut off OF = c.

=

EG || to DF, cutting OQ in G.

Join DF. Through E draw

Then OG is the fourth proportional to a, b, c.

i.e. OG is a fourth proportional to a, b, c.

¶Ex. 10. Find a fourth proportional to a, b, c, using Th. 59.
For Practical Exercises see Practical Geometry, p. 149.

MEAN PROPORTIONAL.

If x is such a quantity that a: x = x: b, then x is called the mean proportional between a and b.

tional between two numbers is the square root of their product.

To find the mean proportional between two given straight lines.

Let a, b be the two given straight lines. Construction Draw a straight line PQ.

From PQ cut off PR = a, and RS = b.

On PS as diameter describe a semicircle.

Through R draw RT to PS to cut the semicircle at T.
Then RT (x) is the mean proportional between a, b.

Proof Since

PTS is the angle in a semicircle and TR is to PS,
.. TR2 = PR. RS,
Th. 60, Cor. 1.

.. x2= a. b.

.. x is the mean proportional between a and b.

NOTE. This construction also solves the problem of constructing a square equal to a given rectangle.

¶Ex. 11. Use Th. 59 Cor. to construct a mean proportional to a, b when both are measured from the same point. Repeat, using Th. 60, Cor. 2.

To construct a figure similar to a given figure and any multiple of its area.

Let A be the area of the given figure and a one side. Let x be the side of the figure to be constructed corresponding to a. Suppose the area of this figure is to be mA.

PROPORTIONAL DIVISION.

Ex. 12. D is a point in the side AB of ▲ ABC; DE is drawn parallel to BC and cuts AC at E; EF is drawn parallel to AB and cuts BC at F. Prove that AD: DB=BF: FC.

Ex. 18. P is a point in the side AB of ▲ ABC; PQ is drawn parallel to BC and cuts AC at Q; CR is drawn parallel to QB and cuts AB produced at R. Prove that AP: AB=AB: AR.

Ex. 14. A straight line AB is divided internally at C; equilateral triangles ACD, CBE are described on the same side of AB; DE and AB produced meet at F. Prove that FB: BC=FC: CA,

Ex. 15. Draw a triangle ABC; in it take a point O, and join OA, OB, OC; in OA take a point A', through A' draw A'B' parallel to AB to cut OB at B', through B' draw B'C' parallel to BC to cut OC at C'. Prove that C'A', CA are parallel.

Ex. 16. A variable line, drawn through a fixed point O, cuts two fixed parallel straight lines at P, Q; prove that OP : OQ is constant.

Ex. 17. O is a fixed point and P moves along a fixed line. OP is divided at Q (internally or externally) in a fixed ratio. Find the locus of Q.

Ex. 18. From a point E in the common base AB of two triangles ACB, ADB, straight lines are drawn parallel to AC, AD, meeting BC, BD at F, G; show that FG is parallel to CD.

Ex. 19. AB, DC are the parallel sides of a trapezium. P, Q are points on AD, BC, so that AP/PD=BQ/QC. Prove that PQ is || to AB and DC. (Use reductio ad absurdum.)

Ex. 20. Prove that the points (0, 0), (2, 1), (5, 2·5) are in a straight line. In what ratio is the line divided?

Ex. 21. O is the mid-point of the median through A of a triangle ABC. BO meets AC in M. Prove that M is a point of trisection of AC.

Ex. 22. A point O is taken on the diagonal AC of a quadrilateral ABCD and the lines OX, OY drawn through O parallel to CB, CD meet AB, AD in X, Y respectively. Prove that XY is parallel to BD.

Ex. 23. A point O is taken within a quadrilateral ABCD, and the lines OX, OY drawn through O parallel to CB, CD meet AB, AD in X, Y respectively. Prove that, if XY is parallel to BD, then O lies on the diagonal AC.

SIMILAR TRIANGLES.

¶Ex. 24. Make a freehand copy of fig. 57 on p. 58, let BC, DE intersect at X. Prove the following pairs of triangles to be equiangular and in each case write down the ratios of the three pairs of corresponding sides :

(i) As XBE, XCD. (ii) As EGF, CBF.

(iii) AS XCD, DHC.

(iv) As DAE, XBE.

(v) As XBE, CBF.

(vi) AS XBE, EGF.

Ex. 25. AB, CD are the parallel sides of a trapezium whose diagonals intersect at O. Prove that As OAB, OCD are similar, and write down the three equal ratios of corresponding sides.

Ex. 26. The altitudes BE, CF of an acute-angled triangle ABC intersect at H. Prove that As HBF, HCE are equiangular and write down the three equal ratios of corresponding sides.

Ex. 27. PQRS is a quadrilateral inscribed in a circle whose diagonals intersect at X; prove that the As XPS, XQR are equiangular. Write down the three equal ratios of corresponding sides.

PQ XP
SR XS

Ex. 28. In the figure of Ex. 27, prove that =

[If you colour PQ, SR red, and XP, XS blue, you will see which two triangles you require.]

Ex. 29. XYZW is a cyclic quadrilateral; XY, WZ produced intersect at a PY PZ point P outside the circle; prove that PWPX

Ex. 30. TP touches a circle at P, TQR cuts it at Q, R; prove that TP TR

Ex. 31. XYZ is a triangle inscribed in a circle, XN is an altitude of the triangle, and XD a diameter of the circle; prove that XY: XD=XN:XZ. Ex. 32. XYZ is a triangle inscribed in a circle; the bisector of LX meets YZ in P, and the circle in Q; prove that XY: XQ=XP: XZ.

Ex. 33. PQRS is a quadrilateral inscribed in a circle; PT is drawn so that SPT = QPR (see fig. 136). Prove that

(i) SP: PR=ST: QR, (ii) SP: PT=SR: TQ. Ex. 34. ABC is a triangle right-angled at A; prove that the altitude AD divides the triangle into two triangles which are similar to ▲ ABC. Write down the ratio properties you obtain from the similarity of ▲s BDA, BAC.

S

Fig. 136.

R

Ex. 35. The altitude QN of a triangle PQR right-angled at Q cuts RP in N; QN PN prove that

=

RN QN

[Find two equiangular triangles; colour the given lines; see Ex. 28.]

Ex. 36. XYZ is a triangle inscribed in a circle, XN is an altitude of the

XY YD
XN NZ'

triangle and XD a diameter of the circle; prove that =

property can be obtained from this by clearing of fractions?

What rectangle

Ex. 37. With the same construction as in Ex. 36, prove that*

XZ. NY=XN. ZD.

[You will have to pick out two equal ratios from two equiangular triangles. If you colour XZ, NY red and XN, ZD blue, you will see which are the triangles.]

* Beware of muddling XZ. NY (which means XZ × NY) and XZ : NY (which means XZ÷NY).

Ex. 38. ABCD is a quadrilateral inscribed in a circle; its diagonals intersect at X. Prove that (i) AX. BC=AD. BX, (ii) AX. XC=BX.XD.

Ex. 39. ABCD is a quadrilateral inscribed in a circle; AB, DC produced intersect at Y. Prove that

(i) YA. BD=YD. CA, (ii) YA.YB=YC.YD.

Ex. 40. The bisector of the angle A of ▲ ABC meets the base in P and the circumcircle in Q. Prove that the rectangle contained by the sides AB, AC is equal to the rect. AP. AQ.

Ex. 41. In Ex. 33, prove that PQ. SR=PR. TQ.

Ex. 42. ABC is a triangle with AB double BC. E is a point in AB such that EB is half BC. Prove that LBCE=LCAB.

Ex. 43. Through the vertex A, of a triangle ABC, DAE is drawn parallel to BC and AD is made equal to AE; CD cuts AB at X and BE cuts AC at Y; prove XY parallel to BC.

Ex. 44. S is a point in the side PQ of ▲ PQR; ST is drawn parallel to QR and of such a length that ST:QR=PS:PQ. Prove that T lies in PR. [Prove 4SPT=4QPR.]

Ex. 45. In a triangle ABC, AD is drawn perpendicular to the base; if BD:DA=DA: DC, prove that ▲ ABC is right-angled.

Ex. 46. The sides AC, BD of two triangles ABC, DBC on the same base BC and between the same parallels meet at E; prove that a parallel to BC through E, meeting AB, CD, is bisected at E.

Ex. 47. Find the locus of a point which moves so that the ratio of its distances from two intersecting straight lines is constant.

Ex. 48. ABCD is any parallelogram. From A a straight line is drawn cutting BC in E and BD in F. Prove that AF: FE=BC: BE.

Ex. 49. ABCD is a rhombus; a straight line through C meets AB and AD, both produced, at P and Q respectively. Prove that PB: DQ=AP2: AQ2.

Ex. 50. A square BCDE is described on the base BC of a triangle ABC, and on the side opposite to A. If AD, AE cut BC in F, G respectively, prove that FG is the base of a square inscribed in the triangle ABC.

Ex. 51. ABC is a triangle, P is any point in AB. Through A on the side of AC away from B draw AE perpendicular to AC and equal to it. Through P draw PM parallel to AC to meet BC in M and draw PK parallel to AE to meet BE in K. Prove that PK and PM are equal.

Ex. 52. Through the vertices B, C of a triangle ABC two parallel lines BL and CM are drawn, meeting any straight line through A in L and M respectively. If LO is drawn parallel to AC and meets BC in O, prove that OM is parallel to AB.

Ex. 58. Prove that two quadrilaterals are similar if the sides and one diagonal of the one are proportional to the corresponding lengths in the other.

S. H. T. G

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