A small circle fits between two touching circles so that all three
circles touch each other and have a common tangent? What is the
exact radius of the smallest circle?
Two semicircle sit on the diameter of a semicircle centre O of
twice their radius. Lines through O divide the perimeter into two
parts. What can you say about the lengths of these two parts?
Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?
Congratulations to Tony from State College Area High School,
Pennsylvania, USA for this solution.
First of all, here is the solution to finding the equation of
the orthogonal circle for the circles with centers of $(0,0)$,
$(3,0)$, $(9,2)$ and radii respectively of $5$, $4$, and $6$.
As the circles are orthogonal we can draw three right angled
triangles. One of the legs of each right triangle is the radius of
one of the given circles, the other leg is the radius of the
unknown orthogonal circle, and the hypotenuse is the distance
between the center of the known circle and the center of the
unknown orthogonal circle.