Given a point $(a,b)$ with $0<b<a$, determine the minimum perimeter of a triangle with one vertex at $(a,b)$, one on the $x$-axis, and one on the line $y=x$.  You may assume that a triangle of minimum perimeter exists.
1. Let the points \( P \), \( C \), and \( D \) be \((a,b)\), \((c,0)\), and \((d,d)\), respectively. We need to find the minimum perimeter of the triangle \( \triangle PCD \).

2. Reflect point \( P \) across the \( x \)-axis to get point \( P' \) with coordinates \((a, -b)\). Reflect point \( P \) across the line \( y = x \) to get point \( P'' \) with coordinates \((b, a)\).

3. The perimeter of \( \triangle PCD \) is given by \( PC + CD + DP \). By reflecting \( P \), we can use the fact that \( PC = P'C \) and \( DP = DP'' \).

4. Therefore, the minimum perimeter is the minimum value of \( PC + CD + DP = P'C + CD + DP'' \).

5. The expression \( P'C + CD + DP'' \) represents the length of a path from \( P' \) to \( P'' \) that passes through the \( x \)-axis and the line \( y = x \). The shortest path is the direct line segment from \( P' \) to \( P'' \).

6. The coordinates of \( P' \) are \((a, -b)\) and the coordinates of \( P'' \) are \((b, a)\). The distance between these two points is given by the Euclidean distance formula:
   \[
   \sqrt{(b - a)^2 + (a + b)^2}
   \]

7. Simplify the expression inside the square root:
   \[
   (b - a)^2 + (a + b)^2 = (b - a)^2 + (a + b)^2 = (b^2 - 2ab + a^2) + (a^2 + 2ab + b^2) = 2a^2 + 2b^2
   \]

8. Therefore, the minimum length is:
   \[
   \sqrt{2a^2 + 2b^2} = \sqrt{2(a^2 + b^2)}
   \]

Conclusion:
\[
\boxed{\sqrt{2(a^2 + b^2)}}
\]