2. On a given line of unlimited length find a point whose distance from a given point is equal to a given finite straight line.

Describe a circle, with the fixed point C as centre, and the given finite line as radius.

Show that either of the points in which it cuts the given line AB satisfies the problem.

Examine the condition that no solution may

exist.

3. To describe an isosceles triangle, each of whose sides shall be double the base.

Produce AB its own length both ways to C and D.
With A and B as centres, describe circles intersecting in

P and Q.

Show that either PAB or QAB is the triangle required.

4. Find a point which is equidistant from three given points.

See § 1, Ex. 5.

Find in what case this problem

has no solution.

F

5. Find a point which is equidistant from three given lines.

See § 7, Exx. 7 and 8, and show that four points I, J, K, L satisfy the condition. Find in what cases the condition is satisfied by (1) two points, (2) no point.

6. ABC, DEF are two triangles; find a point P such that APBC=AABC in area, and ▲PEF= ADEF.

Each of the four points P1, P2, P3, P4 satisfies the conditions

7. Draw a straight line through a given point such that the part intercepted between two given intersecting straight lines shall be bisected at that point.

A

P

Let O be the given point.

Use the method of § 23, Ex. 6 to find the locus of Q, when

QO=OP.

Let this locus meet AC in R. ROS shall be the line required. Prove by Euc. I. 26.

8. O is a given point, and AB, AC are two given straight lines; draw a straight line from O to AB which shall be bisected by AC.

9. Construct an isosceles triangle equal to a given triangle, and standing on the same base.

10. Construct a triangle equal to a given parallelogram, and having an angle equal to a given angle.

11. Construct a parallelogram equal to a given triangle, and having its perimeter equal to that of the triangle.

Bisect BC in D.

Find DE=(AB+AC).

Use Euc. I. 31 and I. 41.

E

12. Given two points and a line, describe a circle such that its centre shall be on the line, and its circumference shall pass through the two points.

See Ex. 1.

13. Find a point at a given distance from a given point, and equally distant from two other given points.

Use § 23, Exx. 1 and 2.

14. Find a point at a given distance from a given point and at a given distance from a given straight line.

Use § 23, Exx. 2 and 3.

Show under what circumstances

there may be 4, 3, 2, 1, or no solutions.

15. Find a point whose distances from two given straight lines are respectively equal to two given finite straight lines. Use § 23, Ex. 3. Compare § 24, Ex 6.

16. Find a point whose distances from two given circles measured along the radii or radii produced are respectively equal to two given straight lines.

17. If n be a whole number, show how to describe an isosceles triangle each of whose sides shall be n times its base.

Use the method of Ex. 3.

18. Describe an isosceles triangle on a given base with sides equal to a given straight line.

19. Find a point in the base of a triangle, or the base produced, equidistant from the two sides.

Use § 23, Ex. 4, and show that in general either of two points satisfies the conditions., When will there be only one point?

20. Draw a line through a given point such that the given point shall be a point of trisection of the part intercepted between two given lines.

Employ the loci found in § 23, Exx. 14 and 16.

21. O is a given point, and AB, AC are two given straight lines. Draw a straight line from O to AC, such that AB shall cut it in a point of trisection.

22. O is a given point, AB, AC are two given straight lines, and n is a given whole number. Draw a straight line POQ meeting AB in P, AC in Q, so that OP =n times OQ.

Employ the locus found in § 23, Ex. 16.

23. O is a point surrounded by a closed curve ABC, and DE is a straight line outside the curve. Draw a straight line from O to meet ABC in P and DE in Q, so that OP=PQ.

Use the method of Ex. 8.

24. Construct a right-angled isosceles triangle equal to a given square.

Compare Ex. 10.

25. Construct a parallelogram equal to a given square standing on the same base and having an angle equal to half a right angle.

26. Construct a rhombus equal to a given parallelogram having each of its sides equal to the longer side of the parallelogram.

Use Euc. I. 35.

27. Construct a rhombus equal to a given triangle, having each of its sides equal to the base of the triangle.

Use the method of the previous Ex., making the altitude of the rhombus half that of the triangle.