Two Diverging Lines With A Point Between Them

Steps

1. Construct a circle k₁ centered at the given point A.
2. Represent two given lines by circular inversion according to the circle k₁.
3. Construct the common tangents of the circles formed by the last step.
4. Represent these tangents by circular inversion according to the circle k₁ to obtain the resulting circles.

Steps

0. Solve for two dissimilar lines p₁ and p₂ and for a point A that lies outside them.
1. Draw the axis of the angle i bisected by the lines from the problem. The centres of the solution circles must lie on this line.
2. Construct a parabola d whose control line is one of the lines (in this case p₁) and whose focus is point A.
3. The intersections of the angle axis i and the parabola d are the centers of the circles we are looking for.
4. Draw the circles with the found centers passing through the point A.

Steps

1. Construct the bisector of the angle which is defined by the two lines from the assignment. Name this angle bisector i. On this bisector must lie the centers of the circles (these circles are the solution to the assignment).
2. Construct a circle and name it c. Tangents of circle c are the lines from the assignment, with the center of circle c on the angle bisector i. We then use circle c as a pattern for homogenous dilation.
3. Construct a line which passes through the intersection of the lines from the assignment and the point A from the assignment. Name this line k. This will give us the patterns of the touchpoints for the homogenous dilation.
4. Construct two lines which pass through the center of circle c and the intersections of circle c with line k.
5. Construct the parallels to both of the lines constructed in step 4 so that they pass through point A. Thus we have projected these two lines through homogenous dilation and therefore we can find the centers of the circles that are the solution to the assignment.
6. Intersections of the angle bisector i and the projected lines from step 5 are the centers of the final circles.
7. Construct the circles with the found centers, passing through the point A.