6. P and Q are the mid-points of the sides AB and CD of the ABCD; show that PD and BQ trisect AC.

7. ABC is a triangle; AC is bisected in M and BM is bisected in N; AN meets BC in P, and MQ is drawn || AP to meet BC in Q; show that BP=PQ=QC.

Use Ex. 1 to show that

MQ bisects PC,

M

and NP bisects BQ.

B

8. A, C are points on the same side of PQ, and B is the midpoint of AC; through A, B, C parallel straight lines are drawn, meeting PQ in D, E, F; show that AD+CF=2 BE.

Draw AGH || PQ.

Use Ex. 1 and Euc. I. 34 to show A

CF-HF=2(BE-GE),

or CF-AD=2(BE-AD); etc. PD

(A Standard Theorem.)

B

E

9. A, C are points on opposite sides of PQ, and B is the midpoint of AC; through A, B, C parallel straight lines are drawn, meeting PQ in D, E, F; show that AD~CF=2BE.

Observe that this may be considered as a particular case of

the previous Ex.

10. ABCD is a parallelogram; BP, CQ, DR are parallel straight lines meeting a straight line through A in P, Q, R; show that CQ=BP DR according as the line does not or does cut the parallelogram.

Use Exx. 8 and 9 to show BP± DR=20S, etc.

11. If a straight line be divided into any number of equal parts, and a series of parallel lines be drawn through the points of division, the intercepts made by these lines on any intersecting straight line shall be equal.

E

B/K G

Draw AKL || FJ.

Use Ex. 1. to show AK KL;

and hence, by Euc. I. 34, FG=GH, etc.

12. ABC is a triangle, and BP, CQ are drawn perpendicular to PQ, a line through A, and D is the mid-point of BC; show that PD=QD.

MA

Draw DM PQ, and use the method of Ex. 11 and Euc. I. 4.

13. If from any point in the base (or base produced) of an isosceles triangle, straight lines be drawn to the sides making equal angles of given magnitude with the base, the sum (or difference) of these lines shall be constant.

R

B

Let PQ, PR, CD make equal angles QPB, RPC, DCB, with BC.

Draw PL || AB.

Use Euc. I. 29, I. 5, and I. 26 to show that
CL=PR;

and deduce that PQ PR=CD.

14. The sum of the perpendiculars drawn from any point within an equilateral triangle to its sides is constant.

Let PQ, PR, PS be the perpendiculars.

Draw LPM || BC; AD and MN || PS; MO || PQ.
Show as in Ex. 13 that

PQ+PR=MO,

PQ+PR+PS MO+MN=AD.

DEFINITION.-The point in which the diagonals of a parallelogram intersect is called the Centre of the parallelogram.

B

15. Any straight line drawn through the centre of a parallelogram and produced to meet the sides is bisected by the centre.

Use § 15, Ex. 1, and the method of § 11, Ex. 1.

16. Any straight line drawn through the centre of a parallelogram bisects the parallelogram.

In above fig. show ▲POD=AQOB, and use Euc. I. 34.

17. The area of any quadrilateral is equal to that of a triangle having two sides and the contained angle equal to the diagonals of the quadrilateral and their contained angle.

S

Draw PBQ, SDR || AC,

and SAP, RCQ || DB.

Use Euc. I. 34 to show that

R ABCD=PQRS=AQRS.

Cor.-Two quadrilaterals are equal when their diagonals are equal and contain equal angles.

18. ABCD is a rectangle, and P is any point on one of its sides; show that a parallelogram PQRS may be inscribed in the rectangle whose sides are parallel to the diagonals of the rectangle, and whose perimeter is equal to the sum of the diagonals.

M

B

R

P

Draw PQ DB, PLS and QMR || AC.
Use Euc. 1-34 to show

PL=QM, PQ=LM

and Euc. I. 29, 6, and 32 to show

PL=LD=LS, etc.

19. ABC is a triangle; and from D, the mid-point of BC, DE and DF are drawn || BA and CA to meet AC and AB in E and F ; show that FE || BC. Use Ex. 1.

20. ABCD is a quadrilateral having AB || DC and BC=AB+CD; show that the bisectors of angles B, C intersect in the midpoint of AD.

Cut off from BC, BF AB, show ▲ AFD a right

=

angle, and use § 1, Ex. 2, and § 16, Ex. 1.

21. If through the vertices of a triangle straight lines be drawn parallel to the opposite sides, these straight lines shall form a triangle equiangular with the given triangle, but of four times its area. Use Euc. I. 34.

22. If one diagonal of a parallelogram be equal to one of its sides, the other diagonal shall be greater than any of its sides. Use Euc. I. 5 and I. 19.

23. AB is a given straight line, and from A a straight line AC (greater than AB) is drawn in any direction and produced its own length to D. DB is joined, and with A as centre and AD as radius a circle is described cutting DB produced in E. From AE AF is cut off equal to AC, and CF is joined. Show that CF bisects AB.

24. If the mid-points of two opposite sides of a parallelogram are also the mid-points of two opposite sides of another parallelogram, then the remaining sides of the two parallelograms are equal and parallel.

Use Euc. I. 33 and I. 34.

25. The sum of the perpendiculars on the sides of a parallelogram from an internal point is constant.

Examine also the case of an external point.

26. E, F are the mid-points of the sides CA, AB of the triangle ABC, and AP is the altitude from A; show that EPF A.

Use § 12, Ex. 1 and Euc. I. 5.

27. The area of a rhombus is equal to half the rectangle contained by its diagonals.

Use the method of Ex. 17.

28. ABCD is a parallelogram and AP, BQ, CR, DS are parallel straight lines, meeting a straight line PS which does not cut the parallelogram; show that AP+CR=BQ+DS.

Find the intersection of the diagonals and use Ex. 8.