Two circles and their common tangents
The position of two circles depends on the distance between centres d and the radii R and r: move a circle and see how many common points and common tangents there are.
- Russia: grade 8
Guide
In the Russian programme this is Grade 8 geometry, chapter "Circle": the tangent to a circle and its properties; the mutual position of two circles appears in harder problems.
Let R ≥ r and d be the distance between centres. The circles are apart when d > R + r (4 common tangents), touch outside when d = R + r (3), cross when R − r < d < R + r (2), touch inside when d = R − r (1), and one lies inside the other when d < R − r (0).
The length of the common external tangent is l = √(d2 − (R − r)2): the radii to the points of contact are perpendicular to the tangent, which gives a right triangle. Lay solutions out as Given, Find, Solution, Answer.
The "why" question asks for your own explanation; then compare it with the model.