2. Produce a given straight line AB to P, so that AB2+BP2=2AP BP. Suppose P found. Then AB2+BP2=2AP BP, .. 2AB2+BP2=AB2+2AP BP =AP2+BP2. [Euc. II. 7. ... AP2=2AB2, from which the Construction easily follows. 3. On a given base AB, construct a ▲ PAB such that Hence we obtain a construction for PN the locus of P. 4. A, B, C are three points in order in the same straight line. Divide AB in P so that APAC=2PB2+PB BC. Suppose P found, then AP.(AP+PB+BC)= PB.(PB+PB+BC), which is evidently satisfied when AP=PB. P 5. ABC is a triangle in which AB>AC; produce BC to P, so that BC CP AB2 - AC2. Suppose P found. Draw AN1BC, and use § 20, Ex. 5, and § 33, Ex. 1, to show that NP=BN. P 6. Construct a rectangle, having given the area and the difference of two adjacent sides. B 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 PBBQ OQ2-OB2, if O be the mid-point of AB, and BC2 OC2-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 difference of their squares. 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 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+PB2=2AP2, Use Euc. II. 7, etc., to show that (AB+AP)2=3AB2. Then use § 39, Ex. 4. 12. P is a point in AB; find a point Q in AB produced, so that AQ2+BQ2=2AP2+2PB2. 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 Bisect AB in O. Join PA, PB, PO. PA2+PB2=2AO2+2PO2, ... 2AO2+2PO2 = CD2; ... PO2CD2-AO2. [§ 37, Ex. 1. 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>AO2, a line equal to PO may be constructed by using § 20, Ex. 2, and § 22, Ex. 7. When CD2=AO2, O is the only point which satisfies the condition. When CD2<AO2, there is no locus. M 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.) Thus P must be a point in a straight line drawn 1 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 C so that CB=2AC, and use § 37, Ex. 10. 4. The base AB and the sum of the sides PA, PB of 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. Let Q be one position of the point. Draw QRLPB, QNLAB. Prove QR QA and PR=PA; hence, show that QB2-QA2 is constant, and use Ex. 2. 5. Construct a triangle, having given the base, the area, and the sum of the squares on the two sides. The vertex of the triangle must lie on the locus found in § 23, Ex. 5, and also on the locus found in Ex. 1. It is, therefore, one of the four points of intersection. 6. Find the locus of the vertex of a ▲ ABC if the sum of the squares on the medians BE, CF is constant. In figure to § 37, Ex. 5, show that BG2+CG2 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 ▲ PAB are given, find the locus of the point in which the bisector of the P meets the straight line through A perpen dicular 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 § 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. |