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Problem 93.
To inscribe an equilateral triangle in a given square ABCD.

1. From C, with radius AC, describe the quadrant AD.
2. From A and D, with the same radius, cut off AE and

DF. .
3. Bisect AF and ED, and through the points of bisection

draw the lines CG, CH, cutting the sides of the square

in G, H. 4. Draw GH; then GCH is the required equilateral triangle,

inscribed in the given square ABCD.

Problem 94. To inscribe an equilateral triangle in a given hexagon, so that its sides are parallel to three sides of the hexagon.

1. Bisect the alternate sides of the given hexagon (Pr. 1) in

the points A, B, and C.

2. Join these points, and an equilateral triangle will be

inscribed in the given hexagon. NOTE.-By joining the three alternate angles of the hexagon, the largest equilateral triangle it will contain will be inscribed.

Problem 95. To inscribe an equilateral triangle in a given regular pentagon ABCDE.

1. From A as centre, with any radius, describe a semicircle

FGH.

.............

2. From F and H, with the same radius, describe arcs cut

ting the semicircle in K and L. 3. From A, draw lines through K and L, meeting the sides

of the pentagon in M and N respectively. 4. Join MN, and AMN will be the required equilateral tri

angle, inscribed in the given pentagon ABCDE.

Problem 96. To inscribe an isosceles triangle within a given square ABCD, having a given base EF.

1. Draw a diagonal BC, and bisect EF in G (Pr. 1).
2. From B, mark off, on the diagonal BC, BH equal to EG

or GF.

3. With H as centre, and HB radius, cut the sides of the

square AB and BD in the points K and L.

4. Join CK, KL, and LC, and an isosceles triangle CKL will

be inscribed within the given square ABCD.

Problem 97.
To inscribe a square within a given circle A.

1. Find the centre of the circle A (Pr. 45).
2. Draw a diameter BC, and bisect it by another diameter

DE.
3. Join BD, DC, CE, and EB; then BDCE is the square

inscribed within the given circle A.

Problem 98.
To inscribe a square within a given triangle ABC.

1. Draw AD, the altitude of the given triangle (Pr. 21).
2. At point C raise a perpendicular CE (Pr. 2), and make

it equal to the base BC.

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3. Draw the line ED, cutting AC in F.
4. From F, let fall a perpendicular FG on the base BC

(Pr. 3); then FG is one side of the required square.
5. From G, mark off the length FG on the base BC in H;

and from H, with the same length, cut AB in K. 6. Join HK, KF; then KFGH is a square inscribed in the

given triangle ABC.

Problem 99.
To inscribe a square within a given rhombus ABCD.

1. Draw the two diagonals AC, BD.
2. Bisect the two angles AOB, COB (Pr. 4) by the lines

EF, GH, cutting the sides of the rhombus in K and L. 3. Join FH, HK, KL, and LF; then FHKL is the required

square, inscribed within the given rhombus ABCD.

Problem 100.

To inscribe a square in a given trapezium ABCD, which has its adjacent pairs of sides equal.

1. Draw a diagonal BD, bisecting the trapezium and the

angle at B.

2. Find the centre of the figure in point E by bisecting

another angle, as at C (Pr. 4). 3. At point E raise a perpendicular to BD, as EF. 4. Bisect the right angles on either side of EF, and produce

the lines of bisection to cut the trapezium in GHKL. 5. Join GK, KH, HL, and LG, and the figure GKHL will

be the required square, inscribed in the given trapezium ABCD.

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