9. To divide a given line, AB, into any number of equal parts.

parallel to AD, to the line AB; these lines will intersect AB equally, and will be of the required number.

10. Upon a given base to construct a triangle, whose other two sides shall be respectively equal to two given lines, any two of the lines, however, must be greater than the third.

Let A and B be the given lines, and CD be the given

A

B

base; any two of them being

greater than the third. It is required to describe upon CD a triangle, whose other two sides shall be equal to A and B.

D

At the centre C, with the distance CE, equal to B, describe a circle; and at the centre D, with the distance DE, equal to A, describe

another circle, intersecting the first in E; join EC and ED. CED shall be the triangle required.

EXAMPLE 1. Construct the following triangles, whose sides are respectively, 30, 40, 50; 25, 75, 55; 120, 130, 140; 100, 20, 90; taken upon any scale of equal parts.

11. To describe a square on a given line AB.

From the point A erect a perpendicular to AB; make AC equal to AB; and through the point C draw CD parallel to AR; make CD equal to AB, and join BD.

ACDB shall be the square

required.

E

D

12. To construct a trapezoid, having its two perpendiculars and its base given.

At the points A and B, of the base AB, erect two perpendiculars AC and BD of the given lengths, and join CD.

ACDB shall be the figure required.

C

B

13. To construct a trapezium, ABCD, having the sides and the diagonal given.

Conceive the trapezium divided into two triangles, ABC and ADC, having the common base AC, which is the given diagonal. Draw the base AC, and upon it, on their respective sides, construct the required triangles (prop. 10) ADC, ABC; having the sides AD, DC; AB, BC, of the required lengths.

B

F

ABCD shall be the trapezium required.

E

A

14. From a given point A, in the circumference, to draw a tangent to the circle ABD, whose centre is C.

Join CA, and make EA perpendicular to AC; EA shall be the tangent required.

B

15. From a given point A, without the circumference, to

draw a tangent to the circle BED.

Join AC, and upon AC describe the circle AECB. Join AB. AB shall be the tangent required; for ABC is a right angle, being in a semicircle, and therefore AB is at right angles to CB, which is the

E

B

radius. Therefore AB is the tangent required.

16. Through the three given points, A, B, D, not in a straight line, to describe a circle.

Join AB and BD, and bisect them; erect perpendiculars till they meet in C; C will be the centre of the circle.

17. To find a third proportional to two given lines.

Let A and B be two given lines. Draw any two unlimited lines, CD and CE, making any angle between them. From CE cut off CF, equal to A; and from CD, CG, equal to B, join GF; again, take CK, equal to

L

B

CG or B, and through K draw KL parallel to GF. CL is a third proportional to A and B, because by similar triangles, CF: CG:: CK or CG: CL; therefore, if CF be double CG, CG will be double CL.

EXAMPLE 1. Find third proportionals to the following lines, viz. 20, 30; 12, 24; 8, 12; 7, 14.

ANSWER. 45, 48, 18, and 28, respectively.

18. To find a fourth proportional to three lines.

Proceed as in the above, but instead of taking CK, equal to CG or B, take CK equal to the third line; then CL in this case also becomes the fourth proportional.

19. To find a mean proportional between two lines A and B.

.. DC: CF:: CF: CG and CF ✔ DC.CG.

EXAMPLE 1. Find mean proportionals between 36 and 100; 25 and 36; and 7 and 28.

ANSWER. 60, 30, and 14, respectively.

20. To divide a given line, AB, into proportional parts.

Through A draw any unlimited line AK, and take AC and CD of the required proportions; join DB, and through C draw CG parallel A

B

to DB. AG and GB, are the parts required.

For, by similar triangles, AC: AD:: AG: AB, and