Problem 107.

To inscribe a circle within any given triangle ABC.

1. Bisect any two of the angles, as B and C (Pr. 4), and let the bisecting lines be produced and meet at D, the centre of the triangle.

2. From D draw a perpendicular DE, to any side of the triangle (Pr. 3).

3. From centre D, with radius DE, inscribe the required circle, which will be tangential to each side of the given triangle ABC.

Problem 108.

To inscribe a circle within a given square ABCD.

1. Draw the diagonals AD, BC, cutting each other in E.

2. From E, draw EF perpendicular to CD (Pr. 3).
3. From E as centre, with radius EF, draw a circle FGHK,
which will be inscribed in the given square ABCD.

Problem 109.

To inscribe a circle in a given rhombus ABCD.

1. Draw the diagonals AC, BD, cutting each other in E.

2. From E, draw a line perpendicular to any side AD, cutting it in F (Pr. 3).

3. From centre E, with radius EF, inscribe the required circle in the given rhombus ABCD.

Problem 110.

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

1. Bisect any two of its adjacent angles ABC, BCD by lines meeting in E (Pr. 4).

2. From point E, draw EF perpendicular to one of the sides AB (Pr. 3).

3. From centre E, with radius EF, inscribe the required circle within the given trapezium ABCD.

Problem 111.

To inscribe a circle in a given quadrant ABC.

1. Bisect the angle at A, the bisecting line cutting the arc in D (Pr. 4).

F

B

2. At point D, draw a tangent to the arc (Pr. 54), meeting one or both of the sides of the angle produced, as AC in E.

3. Bisect the angle at E (Pr. 4), the line of bisection cutting AD in F, the centre. The circle described from F, with radius FA, will be the required circle, inscribed in the given quadrant ABC.

NOTE.

The same method is to be observed in inscribing a circle in any sector of a circle (acute-angled or obtuse-angled).

Problem 112.-The Trefoil.

To inscribe three equal semicircles, having their adjacent diameters equal, within a given equilateral triangle ABC.

1. Bisect the angles at A, B, and C (Pr. 4), and draw the lines of bisection to meet the sides in D, E, F.

2. Join DE, EF, FD; and from G, in EF, draw GH perpendicular to AB (Pr. 3).

3. From G, with GH as radius, describe an arc, meeting EF in K.

4. Draw a line from K parallel to AB, cutting FC in L, and draw LM parallel to FE (Pr. 8).

5. On LM describe an equilateral triangle LMN (Pr. 18), cutting AD in N.

6. On LM, MN, and LN as diameters, describe the three required semicircles within the given equilateral triangle ABC.

Problem 113.

To inscribe three equal semicircles, having adjacent diameters, within a given circle A.

1. Find the centre A of the given circle (Pr. 45).

2. Draw any diameter BC, and the radius AD perpendicular to it..

3. Trisect the arc BD in E and F (Pr. 5).

4. On the other side of D, cut off DG equal to DF, and draw the diameters FÍ, GK.

5. Join EG, cutting the diameter FH in L.

6. From the centre A, at the distance AL, cut off M and N on the diameters KG and BC.

7. Join LM, MN, NL; then LM, MN, and NL are the adjacent diameters of the three required semicircles to be inscribed within the given circle A.

Problem 114-The Quatrefoil.

To inscribe four equal semicircles, having their diameters adjacent, within a given square ABCD, each touching two sides of the square.

1. Draw the diagonals AD, BC, also the diameters EF,

GH.

2. Bisect AG and EC in the points K and L (Pr. 1).

3. Join KL, cutting EF at M.

4. With centre O, and radius OM, mark 'off N, P, Q.

5. Join MN, NP, PQ, QM, which are the diameters on which to describe the four required semicircles within the given square ABCD.