Problem 93.

To inscribe an equilateral triangle in a given square ABCD.

B

C

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 cutting 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 triangle, 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 GE

3. With H as centre, and HB radius, cut the sides of the square AB and BD in the points K and L.

E

K

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.

B

E

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 Craise a perpendicular CE (Pr. 2), and make it equal to the base BC.

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.

and produce the lines of bisection to meet the hexagon in L, M, N, O.

3. Join LN, NM, MO, and OL by straight lines, and the figure LMNO is the required square, inscribed in the given hexagon ABCDEF.

Problem 103.

To inscribe a square within a given quadrant ABC, two of its corners being in the arc.

1. Draw the chord BC, and at one of the extremities, say B, draw BD perpendicular and equal to it (Pr. 2).