Construction.Construct a right angle 0; and take

OE AB and OF CD, and draw EF.

=

Then take GH

R, similar to P.

required.

=

EF, and on GH construct a polygon, The polygon R will be the polygon

R✩P+Q.

Q.E.F.

COR.-In a similar manner we can construct a polygon similar to two given similar polygons and equivalent to their difference.

PROPOSITION XIX.-PROBLEM.

To construct a polygon similar to a given polygon and equivalent to another given polygon. Let P and Q be any two given polygons.

Then it is required to construct a polygon similar to P and equivalent to Q.

Construction.-Construct squares equivalent to P and Q (Pr. XVI. C.); let m and n denote the respective sides of these squares.

Then find EF, a fourth proportional to m, n, and AB, the base of P; and upon EF, homologous to AB, construct a polygon R similar to P.

Then the polygon R will be the polygon required.
Proof. For by construction we have

Hence R is equivalent to Q, and by construction is sim

ilar to P.

PROPOSITION XX.-PROBLEM.

Q.E.F.

To find two straight lines which have the same ratio as the areas of two given polygons.

Construction.-First transform the

polygons into triangles (Prob. XVI.); and then into squares by Problem XI.; we shall then be required to

find two straight lines in the same A

D

B

ratio as two given squares.

Draw two lines AC and CB at right angles with each other; make AC equal to a side of one of the squares, and CB equal to a side of the other square; draw AB and the perpendicular CD.

Proof.-Then, AC2: BC2

= AD: BD.

Th. 25, C. 1.

Hence, AD and BD are in the ratio of the areas of the given polygons.

Q.E.F.

Ex. 23. Find by construction two straight lines which are to each other as two squares whose sides are 3 and 5 inches respectively.

HARMONIC PROPORTION.

17. If a point M is taken on any line AB between A and B, the line is said to be divided inter

nally into the two segments MA and MB.

A

B

18. If a point M' is taken on the prolongation of AB, the line AB is said to be divided externally into the two segments M'A and M'B.

M'

B

19. The given line AB equals the sum of the two internal segments, and the difference of the two external segments. 20. When a line is divided internally and externally into segments having the same ratio, the line is said to be divided harmonically.

PROPOSITION XIX.-PROBLEM.

To divide any line internally and externally in the same ratio; that is, in harmonic proportion. Let AB be the given line.

M

Divide AB into 5 equal parts, and take MA equal to 2 of these parts; then we have

Then divide AB into 2 equal parts, and prolong BA and take the point M' 4 of these parts from A.

COR. 1.—The bisectors of an interior and an exterior angle at the same vertex of a triangle divide the opposite sides harmonically.

and M' divide the line BC harmonically, the points B and C divide the line MM' harmonically.

BM: CM = BM' : CM'.

Cor. 1.

For
By alternation, BM: BM' CM: CM'.

That is, the ratio of the distances of B from M and M' is equal to the ratio of the distances of C from M and M'.

The four points B, C, M, and M' are called harmonic points, and the two pairs B, C and M, M' are called conjugate harmonic points.

REMARK. Three numbers are in harmonical proportion when the first is to the third as the difference of the first and second is to the difference of the second and third. Thus, 2, 3, and 6 are in harmonical proportion, since 2: 6 :: 1: 3, or 6:2=

3:1.

Four numbers are in harmonical proportion when the first is to the fourth as the difference of the first and second is to the difference of the

third and fourth. Thus, 9, 12, 16, and 24 form an harmonical proportion; for 9: 24:: 3: 8, or 24: 98: 3. 14

PROBLEMS FOR SOLUTION.

1. To construct a rectangle, given one side and the diagonal. 2. To construct a square, having given the diagonal of the square. 3. Tɔ divide a triangle into two equal parts by drawing a line from the vertex to the base.

4. To divide a triangle into two parts which are to each other as 2 to 3, by drawing a line from the vertex to the base.

5. To inscribe a circle in a given square. To inscribe a circle in a given rhombus.

6. To construct a square equivalent to the sum of two squares whose sides are 3 and 4 respectively.

7. To construct a square equivalent to the difference of two squares whose sides are respectively 5 and 7.

8. To construct a square that shall be two times a given square. 9. To construct a square that shall be five times a given square. 10. To construct a square so that the vertex of one angle is in the circumference of a circle and two of its sides are tangents.

11. In a given square, to inscribe an equilateral triangle having its vertex in the middle of the side of the square.

12. Given the area and perpendicular of a right triangle, to construct the triangle.

13. Draw two tangents to a given circle which shall contain an angle equal to a given angle.

14. To divide one side of a triangle into two parts proportional to the other two sides.

15. To construct a square equivalent to the sum of a given triangle and a given parallelogram.

16. To construct a square equivalent to the difference between a given parallelogram and a given triangle.

17. To construct a triangle equivalent to the sum of two given triangles. To the difference of two given triangles.

18. To construct a right triangle equivalent to a given triangle. 19. To construct an isosceles triangle equivalent to a given triangle. 20. To construct a square equivalent to an equilateral triangle. 21. To construct a parallelogram whose area and perimeter shall be respectively equal to the area and perimeter of a given triangle.