Point $P$ is located inside a square $ABCD$ of side length $10$. Let $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of $P AB$, $P BC$, $P CD$, and $P DA$, respectively. Given that $P A+P B +P C +P D = 23\sqrt2$ and the area of $O_1O_2O_3O_4$ is $50$, the second largest of the lengths $O_1O_2$, $O_2O_3$, $O_3O_4$, $O_4O_1$ can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.
1. **Understanding the Problem:**
   - We are given a square \(ABCD\) with side length \(10\).
   - Point \(P\) is inside the square.
   - \(O_1, O_2, O_3, O_4\) are the circumcenters of triangles \(PAB, PBC, PCD, PDA\) respectively.
   - We know \(PA + PB + PC + PD = 23\sqrt{2}\).
   - The area of quadrilateral \(O_1O_2O_3O_4\) is \(50\).
   - We need to find the second largest length among \(O_1O_2, O_2O_3, O_3O_4, O_4O_1\) and express it in the form \(\sqrt{\frac{a}{b}}\), where \(a\) and \(b\) are relatively prime positive integers, and then compute \(100a + b\).

2. **Scaling and Cyclic Quadrilateral:**
   - By scaling the quadrilateral \(O_1O_2O_3O_4\) by a factor of \(2\) about \(P\), we obtain a cyclic quadrilateral with circumcenter \(P\).
   - The area of the scaled quadrilateral is \(4 \times 50 = 200\).

3. **Properties of Cyclic Quadrilateral:**
   - For a cyclic quadrilateral with circumcenter \(P\), the sides are chords of the circle centered at \(P\).
   - The area of a cyclic quadrilateral can be given by Brahmagupta's formula:
     \[
     \text{Area} = \sqrt{(s-a)(s-b)(s-c)(s-d)}
     \]
     where \(s\) is the semiperimeter and \(a, b, c, d\) are the side lengths.

4. **Calculating Side Lengths:**
   - Let the side lengths of the cyclic quadrilateral be \(2O_1O_2, 2O_2O_3, 2O_3O_4, 2O_4O_1\).
   - The area of the quadrilateral is \(200\), so:
     \[
     \sqrt{(s-2O_1O_2)(s-2O_2O_3)(s-2O_3O_4)(s-2O_4O_1)} = 200
     \]
   - Given the symmetry and properties of the square, we can assume the lengths are equal or follow a specific pattern.

5. **Finding the Second Largest Length:**
   - Since the quadrilateral is symmetric and scaled, the lengths \(O_1O_2, O_2O_3, O_3O_4, O_4O_1\) are likely to be equal or follow a specific ratio.
   - Given the area and symmetry, we can deduce that the lengths are equal or one pair is slightly longer.
   - Let \(O_1O_2 = O_2O_3 = O_3O_4 = O_4O_1 = x\).
   - The area of the quadrilateral is given by:
     \[
     x^2 = 50 \implies x = \sqrt{50}
     \]
   - The second largest length is also \(\sqrt{50}\).

6. **Expressing in the Form \(\sqrt{\frac{a}{b}}\):**
   - \(\sqrt{50} = \sqrt{\frac{50}{1}}\), where \(a = 50\) and \(b = 1\).

7. **Computing \(100a + b\):**
   - \(100a + b = 100 \times 50 + 1 = 5000 + 1 = 5001\).

The final answer is \(\boxed{5001}\).