Unizor - Geometry2D - Apollonius Problems - Points and Circles

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Notes to a video lecture on http://www.unizor.com

The problems below are easily solvable using the transformation of inversion (symmetry relative to a circle). We recommend to review a previous lecture dedicated to this topic.
The main technique we will use is transformation of a circle passing through a center of an inversion circle into a straight line.
Here is a quick reminder of how inversion works.
An inversion circle with radius R and center O is given. Then any point P (other than center O) is transformed into point P' such that P' lies on a ray from center O to given point P and OP·OP'=R²
We have proven that any circle passing through a center of an inversion circle is transformed by an inversion into a straight line and a circle not passing through a center of inversion is transformed into a circle.

Apollonius Problems -
Points and Circles

Problem PPC
Construct a circle passing through two given points, A and B, and tangential to a given circle c, presuming that both points are located outside of circle c.

Solution
Analysis:
Assume, our circle is constructed. It passes through points A and B and is tangential to circle c.
Using point A as a center of inversion and any radius draw a circle that will serve as an inversion circle.
Transform all elements relative to this inversion circle.
Point B will transform into point B', circle c will be transformed into circle c' and a circle we assumed as constructed (passing through a center of inversion A) will be transformed into a line that will be a tangent from point B' to circle c'.
Construction:
Draw an inversion circle q with center at point A and some radius R.
Construct images of point B and circle c relative to inversion circle q.
Draw a tangent from point B' to circle c'.
Transform a tangent through inversion relative to our inversion circle q. The resulting circle would be tangential to circle c and pass through points A and B.

Problem PCC
Construct a circle passing through a given points, A and tangential to two given circles, c and d, presuming that none of the given elements lies inside the other.

Solution
Analysis:
Assume, our circle is constructed. It passes through point A and is tangential to circles c and d.
Using point A as a center of inversion and any radius draw a circle that will serve as an inversion circle.
Transform all elements relative to this inversion circle.
Circle c will be transformed into circle c', circle d will be transformed into circle d', and a circle we assumed as constructed (passing through a center of inversion A) will be transformed into a line that will be a tangent to both circles c' and d'.
Construction:
Draw an inversion circle q with center at point A and some radius R.
Construct images of circles c and d relative to inversion circle q.
Draw a tangent to circles c' and d'.
Transform a tangent through inversion relative to our inversion circle q. The resulting circle would be tangential to circles c and d.