Question

Three distinct lines are drawn in a plane. Suppose there exist exactly n circles in the plane tangent to all the three lines, then the possible values of n is/are

• A. 0
• B. 1
• C. 2
• D. 4

Answer 1

Answer: (A), (C), (D)

0, 2, 4

Answer 2

The number of circles tangent to three distinct lines depends on their configuration:

1. Three parallel lines: No circle can be tangent to all three. $n=0$.
2. Two parallel lines and one transversal: Two circles are possible, one on each side of the transversal, between the parallel lines. $n=2$.
3. Three lines forming a triangle (no two parallel, not concurrent): There is one incircle and three excircles, totaling four circles. $n=4$.
4. Three concurrent lines: The only point equidistant from all three lines is their point of intersection. A circle centered here would have a radius of 0, which is conventionally not considered a circle. $n=0$.

Thus, the possible values for $n$ are 0, 2, and 4.