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A

B

2. Produce a given straight line AB to P, so that

AB'+BP2=2AP.BP. Suppose P found. Then

AB2+BP2=2AP.BP, :. 2AB+BP2=AB2-+2AP.BP

=AP2+BP2. [Euc. II. 7.

.: AP2=2AB?, from which the Construction easily

follows.
3. On a given base AB, construct a A PAB such that

PA?=PB2+3AB.
Suppose PAB the required triangle,
and draw PN LAB.
PAP=AB2+PB2+2AB BN,

[Euc. II. 12. but PA=ABP+PB2+2AB AB,

[Hypothesis. .:: BN=AB.

Hence we obtain a construction for PN the locus of P.

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N

P

4. A, B, C are three points in order in the same straight line. Divide AB in P so that AP AC=2PB2+PB BO. Suppose P found, then

AP (AP+PB+BC)=
PB (PB+PB+BC),

A B which is evidently satisfied when

AP=PB. 5. ABC is a triangle in which AB>AC; produce BC to P, so that

BC.CP=AB2- AC2.

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Suppose P found. Draw ANIBC, and use § 20, Ex. 5, and $ 33, Ex. 1, to show that NP=BN.

P

A

6. Construct a rectangle, having given the area and the difference of two adjacent sides.

Let AB be the given difference, BC2 the given area.

Produce AB to P and Q, and suppose PB:BQ=BC2. Then PB-BQ=AB.

.:: PA=BQ. Also PB-BQ=OQ-OBY, if O be the mid-point of AB, and BC2=0C2-OB2.

.: OQ=OC. 7. Produce AB to P, so that AB2+AP2=2AP.BP.

Compare Ex. 2. 8. Divide a given straight line externally, so that twice the rectangle contained by the segments may be equal to the differ. ence of their squares.

Here AP2=BP2+2AP.BP.
Write AB+BP for AP and use II. 4, etc.

9. A, B, C are three points in a straight line; find a point P in AC produced such that

BP2= AP.BC.
It will be found that the required condition reduces to

BP.CP=AB.BC. If by Euc. II. 14 a square be formed equal to AB-BC, we have the case of $ 39, Ex. 3.

10. O is the mid-point of AB; divide AB in P so that

4AP.PB=3A02.

11. Divide AB in P, so that AB2+PB:=2AP2.

Use Euc. II. 7, etc., to show that (AB+AP)=3AB2. Then use $ 39, Ex. 4.

12. Pis a point in AB; find a point Q in AB produced, so that AQ2+BQ2=2AP2+2PB?.

13. Produce AB both ways to P and Q, so that PA'AQ and PB:BQ may be respectively equal to two given squares.

$ 41. Loci and intersection of Loci.

1. Find the locus of a point, the sum of the squares on whose distances from two given points is equal to a given square.

(A Standard Locus.)

Let P be any point which satisfies the condition

PA+PB2=CD.
Bisect AB in O. Join PA, PB, PO.

PAP+PB2=2A02+2PO2, [$ 37, Ex. 1. .:: 2A02+2PO2=CD2;

.. PO2=1CD2— AO2.

Thus PO is of constant length, and P must lie in the circumference of the circle whose centre is O and radius PO, and any point in this circumference satisfies the condition.

When ¿CD2>AO?, a line equal to PO may be constructed by using $ 20, Ex. 2, and $ 22, Ex. 7. When CD2=AO?,

O is the only point which satisfies the condition. When ICD?<AO?, there is no locus.

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2. Find the locus of a point, the difference between the squares on whose distances from two given points is equal to a given square.

(A Standard Locus.)

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Let PA2 - PB2=CD2.

Draw PNIAB.
Then PA2 - PB2=AN2 – NB2.
Thus P must be a point in a straight line drawn I AB
through a point N in AB, such that AN2 – NB2=CD2

The point N may be found as in § 27, Ex. 9. There will
evidently be two such points.
3. A and B are two fixed points; find the locus of P, when

2PA2+PB2 is constant.

Divide AB in so that CB=2AC,

and use $ 37, Ex. 10.

A

B

4. The base AB and the sum of the sides PA, PB of a A PAB are given; find the locus of the point in which the bisector of the exterior angle at P meets the straight line through A perpendicular to PA. \R

Let Q be one position of the point.

Draw QRIPB, QNLAB. Prove QR=QA and PR=PA; hence, show that QB-QA2 is constant, and use Ex. 2.

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5. Construct a triangle, having given the base, the area, and the sum of the squares on the two sides.

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6. Find the locus of the vertex of a A ABC if the sum of the squares on the medians BE, CF is constant. In figure to $ 37, Ex. 5, show that BG2+CG? is constant. Use Ex. 1 to show that DG and hence DA is constant. 7. A, B are two fixed points; find the locus of P, when

3PA2+PB2 is constant.

See § 37, Ex. 17.
8. A, B, C are fixed points; find the locus of P, when

PA2+PB2+PC2 is constant.
Use $ 37, Ex. 11. Compare Exx. 1, 3.

§ 9. The base AB and the difference of the sides PA, PB of a A PAB are given, find the locus of the point in which the bisector of the ZP meets the straight line through A perpendicular to PA.

Compare Ex. 4 and use Ex. 2. 10. Construct a triangle, having given the base, the area, and the difference of the squares on the two sides.

Use Ex. 2 and 8 23, Ex. 5. 11. Construct a triangle, having given the base, and the sum and difference of the squares on the two sides.

12. Construct a triangle, having given the base, an angle adjacent to the base, and the sum of the squares on the two sides.

13. Construct a triangle, having given the base, an angle adjacent to the base, and the difference of the squares on the two sides.

14. Construct a right-angled triangle, having given the hypotenuse, and the difference of the squares on the two sides.

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