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"So what happens when a wand meets its brother?" said Sirius.
"They will not work properly against each other," said Dumbledore. "If, however, the owners of the wands force the wands to do battle…a very rare effect will take place. One of the wands will force the other to regurgitate spells it has performed - in reverse.
What exactly determines which wand wins and forces the other?
Initially Voldemort's wand was "winning" as he was magically more powerful than Harry.
His wand began to vibrate more powerfully than ever . . . and now the beam between him and Voldemort changed too ... it was as though large beads of light were sliding up and down the thread connecting the wands - Harry felt his wand give a shudder under his hand as the light beads began to slide slowly and steadily his way. . . . The direction of the beams movement was now toward him, from Voldemort, and he felt his wand shudder angrily. . . .
Goblet of Fire. Chapter 34, Priori Incantatem
But Harry's courage in the face of death ultimately gave him power over Voldemort.
He concentrated every last particle of his mind upon forcing the bead back toward Voldemort, his ears full of phoenix song, his eyes furious, fixed . . . and slowly, very slowly, the beads quivered to a halt, and then, just as slowly, they began to move the other way . . . and it was Voldemort's wand that was vibrating extra-hard now . . . Voldemort who looked astonished, and almost fearful. . . .
Voldemort biggest weakness was his fear of death which is what made him weaker than Harry on that day. This has also been explained by Dumbledore to Harry in the last book.
“Having ensured this two-fold connection, having wrapped your destinies together more securely than ever two wizards were joined in history, Voldemort proceeded to attack you with a wand that shared a core with yours. And now something very strange happened, as we know. The cores reacted in a way that Lord Voldemort, who never knew that your wand was a twin of his, had never expected.
“He was more afraid than you were that night, Harry. You had accepted, even embraced, the possibility of death, something Lord Voldemort has never been able to do. Your courage won, your wand overpowered his. And in doing so, something happened between those wands, something that echoed the relationship between their masters.
Deathly Hallows, Chapter 35, Kings Cross
