A large panel needs to be moved through a corridor, the panel is tall as the corridor. The corridor is A feet wide before a right angle turn, after the turn, it is B feet wide. What is the maximum length of the panel that can pass through this corner.

Overhead view of the hallway:

+------------+---
|           /   |
|          /    |B ft
|         /     |
|        /+------
|       / |
|      /  |
|     /   |
|    /    |
|   /     |
|  /      |
| /       |
|/        |
+<-A ft-->|

In order to bypass the corner, at the moment the panel endpoints touch the walls, the portion of the panel at the a-wide corridor should have the same length of the portion at the b-wide corridor (I can't imagine how to prove this, but it's intuitive to me).

At this moment, the panel, the walls and the corridors space form two right triangles with hypotenusa L. The other sides of each triangle will be a and b (the triangles are similar by construction). So, the panel length is 2*L = 2*(a^2 + b^2).