Now we solve the problem of Apollonius to construct a circle tangent to three given, based on the concept of an orthogonal imaginary circle, discussed in the previous lesson.

Start a new project.

Let three circles sre given the plane, which are clearly disjoint.

Fig. 1

Construct tangents o1 and o2 to circles d1 and d3.

Fig. 2

Find the intersection point p5 of the tangents o1 and o2.

Fig. 3

Construct the intersection point p10 tangent to circles d2 and d3.

Fig. 4

Construct the intersection point p15 tangent to circles d1 and d3.

Fig. 5

Draw a straight line o7 passing through the three points p5, p10 and p15. To do this, select the point and press the button with the Latin symbol q.

Fig. 6

Find the radical center of three circles - point p16.

Fig. 7

Determine the center of the original circle - points p17, p18, p19.

Fig. 8

Construct triangle with sides o10, o11, o12 based on points p17, p18, p19

Fig. 9

Map the radical center p16 in the inversion circles d1, d2 and d3. We get the points p20, p21, p22.

Fig. 10

Draw circles d4, d5, d6 passing trough the radical center and all possible pairs of a triad of points p20, p21, p22 .

Fig. 11

Find the points p23 and p24 of intersection of the line o12 and d5 circle.

Fig. 12

Calculate the radius of the imaginary circle orthogonal finding the distance c1 between the radical center point p16 and the point p24.

Fig. 13

Define an imaginary circle d7 with real center of p16 and the value c1.

Fig. 14

Points p25 and p26 are the intersection of circles d7 and d1.

Fig. 15

Draw a straight line o13 passing through the points p25 and p26.

Fig. 16

Points p27 and p28 are the intersection of circles d7 and d2.

Fig. 17

Draw a straight line o14 passing through the points p27 and p28.

Fig. 18

Points p29 and p30 are the intersection of circles and d7 d3.

Fig. 19

Draw a straight line o15 passing through the points p29 and p30.

Fig. 20

Construct a circle d8, passing perpendicular to the lines o13, o7 and circle d1.

Fig. 21

Find the intersection points p31, p32 of circles d8 and d1.

Fig. 22

Construct a circle d9, passing perpendicular to the lines o15, o7, and circle d3 .

Fig. 23

Find the intersection points p33, p34 of circles d3 and d9.

Fig. 24

Construct a circle d10, passing perpendicular to the line o14, o7 and circle d2.

Fig. 25

Find the intersection points p35, p36 of circles d10 and d2.

Fig. 26

Draw a circle d10, passing orthogonal to the lines o7, o14 and circle d2.

Fig. 27

Draw the desired d11 circle passing through the points p32, p34, p35 and d12 passing through the points p31, p36 and p33.

Fig. 28