Problem 151.

To inscribe a triangle in a given circle A, similar to a given triangle BCD.

1. Draw a tangent EFG at any point F in the circumference (Pr. 54).

2. From F, draw FH, making with EF an angle equal to BCD (Pr. 10), and meeting the circumference in H.

3. From F, draw FK, making with FG an angle equal to BDC, and meeting the circumference in K.

4. Join HK. Then FHK is a triangle similar to the given triangle BCD, inscribed within the given circle A.

Problem 152.

To construct a triangle similar to a given triangle A, and having its perimeter equal to a given straight line BC.

1. On BC, construct a triangle BDC, having its angles equal to those of the given triangle A (Pr. 23).

2. Bisect the angles at B and C (Pr. 4) by lines meeting in E.

3. Through E, draw EF and EG parallel to DB and DC (Pr. 9), meeting BC in F and G. Then EFG is the required similar triangle, having its perimeter EF, FG, GE equal to the given straight line BC.

Problem 153.

To inscribe a triangle within a given triangle ABC, and similar to another given triangle DEF.

1. On the side AC of the triangle ABC, construct a triangle ACG similar to the given triangle DEF by measuring the angles at E and F.

2. Join BG, cutting AC in H.

3. Through H, draw HK parallel to AG (Pr. 9), and HL parallel to GC, meeting BC in L.

4. Join KL. Then HKL will be the required triangle inscribed within the given triangle ABC, and similar to the given triangle DEF.

Problem 154.

To construct within a given square ABCD, another square concentric with it, and having its side equal to a given line E.

1. Draw the diagonals AC, BD; and on AB cut off AF equal to E.

2. Through F, draw FG parallel to AC (Pr. 9), and through G, draw GH parallel to AB.

3. Through G and H, draw GK and HL parallel to BC or AD.

4. Join LK. Then GHLK is the required concentric square, and having its side equal to the given line E.

NOTE. If it is required to describe a square about a given square, make AB produced equal to the required side, and proceed as in Problem 150.

Problem 155.

To construct a rectangle, similar to a given rectangle ABCD, on a part ED of the side CD of the given rectangle.

1. Draw the diagonal AD, and from E, draw a line EF parallel to AC or BD (Pr. 9), meeting AD in the point F.

2. From F, draw a line FG parallel to CD (Pr. 9), meeting BD in the point G. Then DEFG is the required rectangle, and is similar to the given rectangle ABCD.

Problem 156.

To construct a trapezium on a given line AB, which shall be similar to a given trapezium CDEF.

1. At B, in the given line AB, make angle ABG equal to angle CDF (Pr. 10), and angle ABH equal to angle CDE.

F

E

2. Make angle BAK equal to angle DCF.

3. At K, make angle BKL equal to angle DFE. Then AKLB will be the required trapezium, constructed on the given line AB, and similar to the given trapezium CDEF.

NOTE. By means of this Problem, any rectilineal figure may be constructed similar to another given rectilineal figure, either greater or less.

SECTION XI.-EQUIVALENT AREAS.

(Before the student enters on the following problems he should thoroughly master the subjoined theorems.)

(A.) "Parallelograms upon the same base, and between the same parallels, are equal to each other" (in area).—Euc. I., 35. Ex. ABCD, DBCF

F

B

NOTE 1.-"Between the same parallels means having the same altitude.

NOTE 2.-The altitude must always be perpendicular to the base. Ex. AB

A

B

(B.) "Parallelograms upon equal bases and between the same parallels