Two Circles With Inner Contact, Point Inside The Circles

Number of solutions: 1

SOLUTION METHOD

Circle Inversion

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Steps

1. We choose the reference circle for the inversion. We take the point of tangency of the two circles as its center.
2. We apply a circular inversion to the given objects. Since both circles pass through the center of the reference circle, they are mapped to lines.
3. The image of the solution circle is a line that is parallel to the images of the circles and passes through the image of the point.
4. We invert the obtained line back using the original inversion.
5. The problem has one solution.

SOLUTION METHOD

Loci of Points Satisfying Certain Properties

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Steps

0. Given are the circles a and b, which are touching internally. The point of tangency between them is D. Also given is the point A inside both circles.
1. Draw the line c that goes through the centres of a and b and intersects them in the point D.
2. Construct the perpendicular bisector of AD. It intersects c in the point S.
3. Draw the circle k that originates in S and has |SA| as its radius. It is the only solution.

SOLUTION METHOD

Circle Inversion

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Steps

1. Draw a circle e centred at A and passing through S0
2. Represent the given circles and the given point A through the circular inversion. This will be displayed at infinity
3. Draw the tangent of the circles
4. We display this tangent back through the circular inversion