7. The ZACB of a A ABC is four-thirds of a right angle; show that AB2= AC2+BC2+AC BC.

8. The ZACB of a A ABC is two-thirds of a right angle; show that AB2= AC2+BC2 - AC.BC.

Use $ 13, Ex. 1, and Euc. II. 13. 9. If, in the A ABC, AC=BC, and < ACB=four-thirds of a right angle; show that AB2=3AC2. 10. O is a point within the A ABC such that

_ BOC= _COA= LAOB; show that BC2+CA2+AB2=2(0A2+OB2+0C2)+OB:OC+OCOA+OA.OB. 11. If, in the A ABC, BQ and CR are the altitudes from E and C, and the 28 B, C are both acute; show that

BC2=BA.BR+CA.CQ.

Use Euc. II, 12 or 13, and Ex. (1). Consider the case in which the < B is obtuse.

12. H is the orthocentre of the acute-angled A ABC and AP, BQ, CR are the altitudes ; show that

(1) AH.HP=BH.HQ=CH.HR.

(2) AH+BCP=BHP+CA2=CH2+AB?. For (1) use Ex. 1, for (2) use § 20, Ex. 5 or Euc. II. 12.

13. In the A ABC, AC=BC, and AN I BC, or BC produced ; show that AB=2BC-BN.

Projections. 14. If two straight lines be equal and parallel, their projections on any other straight line shall be equal.

Enunciate and prove two converses to this theorem. 15. A, B are points on the bisector of the < XOY; P, Q are their projections on OX; and R, S are their projections on OY; show that PR QS.

NOTE. — The projection of a point on a straight line is the foot of the perpendicular drawn from the point to the line.

CHAPTER II.

PROBLEM S.

$ 39. Problems which follow directly from known propositions.

1. Divide a given straight line into two parts so that the rectangle contained by the parts shall be equal to a given square.

This is a converse problem to Euc. II. 14, and the following construction is at once suggested.

Construction,

Bisect AB in O, and describe a circle with O as centre and OA as radius.

Through o draw OCIAB and equal to a side of the given square.

Through C draw CP | AB, meeting the circle in P, and through P draw PQLAB.

Q shall be the required point of section.
Proof
-
AQ.QB+OQ?=AO2;

[Euc. II. 5. =PO2;

=PQ2+0Q2; [Euc. I. 47. ..AQ.QB=PQ2.

Observe that if PC be produced to meet the circle again in P', and P'Q' be drawn I AB, a second point will be found to satisfy the condition, but that, if OC >OA, no point satisfies the condition.

Observe, also, that APB is a right-angled triangle, and compare § 32, Ex. 1.

2. Construct a square equal to n times a given square.

a

Let AB be a side of the given square.
Produce AB to C, so that

BC=nAB,
ᄒR

and proceed as in Euc. II. 14. This problem may also be solved by means of Euc. I. 47.

A

B

с

3. Construct a rectangle equal to a given square and having one side equal to a given straight line.

B

Let BC be a side of the given
square and be I AB the given
straight line.
Produce AB and draw CDLAC, etc.

Use $ 32, Ex. 1.
Compare Euc. II. 14.

4. Produce a given straight line, so that the rectangle contained by the whole line thus produced, and the part produced, shall be equal to a given square.

Let AB be the given straight line, CB a side of the given square.

Produce AB and make OP=OC where O is mid-point of AB.

Use Euc. II. 6.
Compare Euc. II. 14.

A

5. Produce a given straight line AB to P, so that ABAP shall be equal to the square on a given straight line AC.

See $ 32, Ex. 1 (2).

B

6. A, B are fixed points and XY is a given straight line; find a point P in XY, such that AP2+BP2 may be a minimum.

B

X

>

Bisect AB in O, and draw OPIXY.

Use $ 37, Ex. 1, and $ 8, Ex. 1. 7. Construct a rectangle having given the area and the perimeter.

Observe that in Ex. 1, AB is one-half the perimeter. 8. Construct a square which shall be equal to

(1) A given parallelogram.

(2) A given triangle. 9. Construct a rectangle equal to the sum of two given squares, so that one side of the rectangle may be equal to a given straight line.

10. Construct a rectangle equal to the difference of two given squares, so that one side of the rectangle may be equal to a given straight line. 11. Divide AB in P, so that

(1) AB2+PB2=3 AP2.

(2) AP2 - PB2= AP-PB. See $ 36, Ex. 2. Can P be a point of external section ?

12. Produce AB to P so that AB:BP may be equal to a given square.

See $ 32, Ex. 1. 13. Divide a straight line so that the rectangle contained by the segments may be a maximum.

Use Euc. II. 5. 14. Divide a straight line so that the sum of the squares on the segments may be a minimum.

Use Euc. II. 9.

$ 40. Analysis and Synthesis.

1. Divide one side of a triangle, so that the sum of the squares on its segments may be equal to the sum of the squares on the other two sides of the triangle.

Analysis.
Suppose that BC is divided in P, so that

AB2+ ACP= PB2+PC.
If D be the mid-point of BC,

AB2+AC?=2AD2+2 BD2. [S 37, Ex. 1.
Also PB+PC=2PD+2BD. Euc. II. 9, 10.
Hence we see that PD=AD.
This suggests the following Construction-

Bisect BC in D.
Join AD and cut off DP=DA.

P is the point required.
Proof
-
AB2+AC2=2AD2+2BD2.

[S 37, Ex. 1. =2PD2+2BD

[Construction. PB2+ PC

[Euc. II. 9, 10. A second solution can be obtained by cutting off

DP'=AD Observe that P is in BC, or BC produced, according as LA is obtuse or acute, and falls on B when _ A is right. Compare $ 12, Ex. 6.

L