To solve this problem, we can use the concept of the tangent to a circle.
Let's assume that the centers of the three circles are O₁, O₂, and O₃, and their radii are r₁, r₂, and r₃, respectively.
Given that the circles touch each other externally, the tangents drawn at the points of contact will pass through the centers of the circles. Therefore, the line joining the centers of two tangent circles will form a right triangle with the radii as the hypotenuse.
Using this concept, we can solve the problem using the given information.
From the given information, we have:
xy = 15
yz = 10
xz = 21
Let's consider the right triangle formed by O₁, O₂, and the point of contact.
In this triangle, the hypotenuse is r₁ + r₂, and the legs are r₁ and r₂.
Using the Pythagorean theorem, we can write the equation as:
(r₁ + r₂)² = r₁² + r₂²
Expanding the equation, we get:
r₁² + 2r₁r₂ + r₂² = r₁² + r₂²
Simplifying, we get:
2r₁r₂ = 0
Since r₁ and r₂ are both positive, this implies that r₁ = 0, which is not possible.
Therefore, the only possible solution is that r₁ = 0, and r₂ + r₃ = xz = 21.
From xy = 15, we can find that r₂ = 15/y = 15/3 = 5.
Substituting r₂ = 5 into r₂ + r₃ = 21, we get:
5 + r₃ = 21
r₃ = 21 - 5
r₃ = 16
Hence, the radii of the three circles are r₁ = 0, r₂ = 5, and r₃ = 16.
Therefore, the correct answer is option A) 0, 5, 16.