Among all triangles having (i) a fixed angle $A$ and (ii) an inscribed circle of fixed radius $r$, determine which triangle has the least minimum perimeter.
1. **Given Conditions and Notations:**
   - We are given a set of triangles $\triangle ABC$ with a fixed inradius $r$ and a fixed angle $\angle BAC = \alpha$.
   - We use standard notations: $a, b, c$ for the sides opposite to vertices $A, B, C$ respectively, $s$ for the semi-perimeter, and $h_a$ for the altitude from $A$ to $BC$.

2. **Area and Semi-Perimeter Relationship:**
   - The area of $\triangle ABC$ can be expressed in two ways:
     \[
     [\triangle ABC] = \frac{a \cdot h_a}{2} = r \cdot s
     \]
   - Solving for $s$:
     \[
     s = \frac{a \cdot h_a}{2r}
     \]

3. **Expression for Semi-Perimeter:**
   - Using the fact that $s = \frac{a \cdot h_a}{2r}$, we need to express $a$ and $h_a$ in terms of $r$ and $\alpha$.
   - From the inradius formula, we know:
     \[
     s = \frac{a + b + c}{2}
     \]
   - We also know that:
     \[
     s - a = \frac{b + c - a}{2}
     \]

4. **Using Trigonometric Relationships:**
   - Using the formula for the inradius in terms of the angle $\alpha$:
     \[
     r = \frac{a \cdot \sin \alpha}{2 \cdot \cot \frac{\alpha}{2}}
     \]
   - Therefore:
     \[
     a = 2r \cdot \cot \frac{\alpha}{2}
     \]

5. **Minimizing the Perimeter:**
   - The perimeter $2s$ is minimized when $h_a$ is maximized.
   - To maximize $h_a$, we consider the geometric properties of the triangle. If the incircle $(I)$ touches $BC$ at $X$ and $\overrightarrow{AI}$ cuts $(I)$ at $D$, then:
     \[
     AD = AI + IX = r + r \csc \frac{\alpha}{2}
     \]
   - Since $AD \ge AX \ge h_a$, the equality holds when $D$ coincides with the foot of the altitude from $A$ to $BC$.

6. **Conclusion:**
   - The triangle $\triangle ABC$ with the least perimeter given a fixed inradius $r$ and a fixed angle $\alpha$ is an isosceles triangle with $A$ as the apex and the base angles equal. This configuration ensures that $h_a$ is maximized, thus minimizing the perimeter.

\[
\boxed{\text{The triangle with the least perimeter is an isosceles triangle with apex angle } \alpha.}
\]