Kissing circles inside a larger circle: Two unit-radius circles touch each other and both touch internally a larger circle of radius 2. Find the radius of a circle that touches all three (the two unit circles and the larger circle).

Introduction / Context:
This is a tangent-circle problem with symmetry. Two unit circles lie inside a larger circle of radius 2, each touching the large circle internally and also touching each other. We must find the radius r of a circle tangent to all three.

Given Data / Assumptions:

• Larger circle radius R = 2 with center at O.
• Two smaller circles have radius 1 and centers at (±1, 0); they are distance 1 from O (since 2 − 1 = 1) and are mutually tangent (distance 2 apart).
• Unknown circle of radius r has center at (0, y) by symmetry (equidistant from the two unit centers).

Concept / Approach:
Use geometric distance conditions for tangency: (i) distance from the unknown center to O is R − r (internal tangency), and (ii) distance from the unknown center to either unit center is r + 1 (external tangency). Solve simultaneously.

Step-by-Step Solution:
Distance to O: √(0^2 + y^2) = y = R − r = 2 − r.Distance to (1, 0) or (−1, 0): √(1^2 + y^2) = r + 1.Thus √(1 + y^2) = r + 1 and y = 2 − r.Substitute: √(1 + (2 − r)^2) = r + 1.Square both sides: 1 + (2 − r)^2 = (r + 1)^2 ⇒ 1 + 4 − 4r + r^2 = r^2 + 2r + 1.Simplify: 5 − 4r = 2r + 1 ⇒ 6r = 4 ⇒ r = 2/3.

Verification / Alternative check:
y = 2 − r = 2 − 2/3 = 4/3. Then √(1 + y^2) = √(1 + 16/9) = √(25/9) = 5/3, and r + 1 = 2/3 + 1 = 5/3, consistent.

Why Other Options Are Wrong:
3/2 is too large (would not fit); 2/5 and 1/3 are too small; 5 is impossible inside R = 2.

Common Pitfalls:
Forgetting that internal tangency uses distance = R − r, not R + r; assuming the center lies on the diameter line instead of off-axis symmetry.

Final Answer:
2/3