POINT • CIRCLE • CIRCLE
Download GeoGebra file
Download GeoGebra file
Two Tangent Circles, Different Sizes, Point Outside A Circle
Number of solutions: 3
GeoGebra construction
Steps
- Given are circles c and d that intersect each other, and a point A.
- Construct circle e with the center A so that both given circles lie inside the constructed circle and do not intersect it.
- Use circle e as the axis of a circle inversion. Circles c' and d' are created.
- Construct lines g and f that are tangents of circles c' and d'.
- Use circle e as the axis of a circle inversion again. By that, circles k1 and k2 are created.
- Solutions are circles k1 and k2.
GeoGebra construction
Steps
- We will solve this problem using a set of points of given properties, which for a pair of a point and a circle represents a hyperbola. Draw the lines through point C and the centers of the given circles from the problem, i.e., points B and A. Label the intersections of the lines and circles from the problem.
- Find the midpoints between the intersections and point C, these points will lie on the hyperbolas we are looking for.
- Using the two foci (the center of the given circle and point C from the assignment) and the point lying on the search hyperbola, one of the centers from the previous step, draw hyperbolas for the two circles from the assignment and point C.
- The intersections of these hyperbolas are the centers of the circles we are looking for, denote them S1, S2 and S3.
- Draw the resulting circles centered at S1, S2, and S3, which pass through point C from the problem.
- Resulting circles of the circle, circle, point problem.