Points Inside A Circle Equidistant From The Centre

Steps

0. We have a circle c and points C and D which lie inside the circle at the same distance from the centre.
1. Draw the line f, which is the axis of the line segment CD.
2. Name the points at the intersection of the circle c and the line f E and F.
3. Draw the lines h and g, the line h being the axis of the line segment CE and the line g being the axis of the line segment CF.
4. At the intersection of the lines g and f lies the point G, and at the intersection of the lines h and f lies the point H.
5. Draw the circles k₁ and k₂. The circle k₁ is centered at point H, passes through points C and D, and the point of contact with the circle c is point E. The circle k₂ is centered at point G, passes through points C and D, and the point of contact with the circle c is point F.

Steps

1. Construct a helper circle with center at the point B such that the point A lies on it.
2. Using circular inversion reflect the circle k about the helper circle.
3. Construct both tangets of the newly drawn circle that intersect at the point A.
4. Reflect these tangets using circular inversion about the helper circle.
5. The resulting circles k₁ and k₂ are the solutions to this problem.