Circle PPC

1. Begin with two points $$P_1$$ and $$P_2$$ and circle centered on A.

2. Construct the line $$\overset{\longleftrightarrow}{P_1 P_2}$$ and the perpendicular bisector of $$\overline{P_1 P_2}$$.

3. Construct an arbitrary circle that is centered on a point B on the perpendicular bisector of $$\overline{P_1 P_2}$$, which passes through $$P_1$$ and passes through the circle A. It intersects the circle at points C and D.

4. Construct the line $$\overset{\longleftrightarrow}{C D}$$. It intersects $$\overset{\longleftrightarrow}{P_1 P_2}$$ at point E.

5. Find the two lines which pass through point E and are tangent to circle A. The points of tangency are $$T_1$$ and $$T_2$$.

6. Construct the lines $$\overset{\longleftrightarrow}{A T_1}$$ and $$\overset{\longleftrightarrow}{A T_2}$$. They intersect the perpendicular bisector of $$\overline{P_1 P_2}$$ at points $$O_1$$ and $$O_2$$.

7. The points $$O_1$$ and $$O_2$$ are the centers of the two circles that pass through $$P_1$$ and $$P_2$$ and are tangent circle A.