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To inscribe a triangle in a given circle 4, similar to a given triangle BCD.
1. Draw a tangent EFG at any point F in the circumfer
ence (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.
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 BÈ.
To inscribe a triangle within a given triangle ABC, and similar to another given triangle DEF.
C 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
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.
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
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.
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 CĎE.
2. Make angle BAK equal to angle DCF.
AKLB will be the required trapezium, constructed on the
CDEF. NOTE.—By means of this Problem, any rectilineal figure may be constructed similar to another given rectilineal figure, either greater or less.
Section X1.-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. Ec. ABCD, DBCF
NOTE 1.-"Between the same parallels" means having the same altitude.
NOTE 2.—The altitude must always be perpendicular to the base. Ex. AB
(B.) “Parallelograms upon equal bases and between the same parallels