The "hinge" is one label for that moment on a story's arc represented by the hyphen in 'problem-solution'. Until the hinge is reached, the action is all building conflict, after it the action is discovering and applying the solution.
Sometimes the hinge and the "mirror moment" described by James Scott Bell in Write Your Novel from the Middle are the same.
Bell writes that the "mirror moment" is usually smack in the middle of the book, and sometimes involves an actual mirror. In this scene, the protagonist confronts his own nature, and solidifies his own sense of his identity. This perceptive insight frees the protagonist to pass the hinge.
Self-discovery may help the character decide to apply a solution he has discovered. It may lead him to search for a different answer to his problem. It might even provide a complete turn-around to the story, as he replaces the first problem identified with a completely different conflict.
This is why Bell recommends that writing the mirror/middle scene can make writing the rest of the novel easier.
For the past week, I have been reading for the first time the quintet of novels by Orson Scott Card that begins with Ender's Game. In that first novel, I was amazed to find an obvious mirror scene, in the exact middle of the novel. Eight-year-old Ender Wiggin has been training in an intently-focused way to command starships in the battle against the distant enemy alien Buggers, when they attack again.
In his free time, Ender and all the other boys he is competing against play the Fantasy Game, This video game is supposed to end with a scenario called the Giant's Drink, an unwinnable contest which measures the boys' willingness to keep trying. (In that, this is like Star Trek's Kobayashi Maru simulation, which James T. Kirk notoriously cheats his way through at Starfleet Academy.)
Ender surprises the designers of the Fantasy Game, who are observing, when he finds a way to win against the Giant, and the game immediately takes him to another hard-to-win scenario that involves a mirror. What he sees in the mirror is not himself, but his older brother Peter, and this informs not only the rest of Ender's Game, but also has repercussions through at least the next three novels as well.
In plotting Roger and the Meteor Mass, I have planned three different hinges. One is a mirror moment, in which the mirror is Roger's perception of himself through the eyes of his mother as they both watch his baby sister pretend to be a cat. While it is not written word-for-word, I have "watched" this scene in my mind's eye many rimes now, and have figured out what insight it gives Roger about his oddness.
The fact that I am past the "half-way" mark in word count, but still haven't reached this moment in my novel, tells me that this book (at least in its first draft) will be longer than 50,000 words.
(Written November 12, but not posted until November 16, 2015.)
Word Count: 2477 Day 11 and 26912 Total