Both Points In A Circle, One Is The Centre

Steps

0. Given is the circle c, inside of which lie the points A and B. A lies in the centre of c
1. Construct the perpendicular bisector of AB, f. The resulting circles' origins will lie on it.
2. Construct a line parallel to f that goes through the point B, g. The points where g intersects c shall be named C and D
3. Construct the perpendicular bisectors of BC (h) and BD (i). The points where they intersect f shall be named F and E, respectively.
4. Draw the circles d and e: d has E as its origin, goes through the points A and B, the point of tangency with c is D e has F as its origin, goes through the points A and B, the point of tangency with c is C

Steps

0. Solve for the circle k and two points A and B, one of which lies inside the circle and the other is its centre.
1. Project the point inside the circle using circular inversion outside the circle.
2. Draw tangents to the circle k passing through the projected point.
3. Using circular inversion on the circle k, project the two tangents inside the circle. The circles k₁ and k₂ shown are solutions to the problem.