The problem involves a triangle $\triangle ABC$ where the sides are given as $AB = 13$, $BC = 14$, and $CA = 15$. A circle passes through point $A$ and is tangent to side $BC$. We are asked to find the smallest possible area of the circle, expressed in terms of $\pi$.

Approach:

This is a classic case of an excircle problem, where the circle is tangent to side $BC$ and passes through a vertex of the triangle. The smallest possible circle that can satisfy these conditions is the incircle of the triangle.

Steps to Solve:

1. Semiperimeter Calculation: The semiperimeter $s$ of triangle $ABC$ is given by: $s = \frac{AB + BC + CA}{2} = \frac{13 + 14 + 15}{2} = 21$

2. Area of Triangle (Heron's Formula): Using Heron's formula, the area $A$ of the triangle can be calculated as: $A = \sqrt{s(s - AB)(s - BC)(s - CA)} = \sqrt{21(21 - 13)(21 - 14)(21 - 15)}$ $A = \sqrt{21 \times 8 \times 7 \times 6} = \sqrt{7056} = 84$

3. Inradius Formula: The radius $r$ of the incircle can be calculated as: $r = \frac{A}{s} = \frac{84}{21} = 4$

Thus, the radius of the smallest circle that passes through $A$ and is tangent to side $BC$ is 4 units. Therefore, the area of the circle is: $\text{Area of the circle} = \pi r^2 = \pi \times 4^2 = 16\pi$

Final Answer:

The smallest possible area of the circle is $16\pi$.

Would you like a more detailed explanation or have any questions?

Here are five related questions to deepen understanding:

1. How is the semiperimeter of a triangle used in calculating its incircle?
2. What is Heron’s formula, and how does it help in finding the area of a triangle?
3. How does the radius of the incircle relate to the area and semiperimeter of a triangle?
4. Can you explain the steps for finding the inradius in a different triangle?
5. What are the differences between an incircle and an excircle in a triangle?

Tip: When solving geometry problems involving circles and triangles, always check whether the circle is an incircle, circumcircle, or an excircle to apply the appropriate formulas.