2.4.10: Tigraine = Shaiel, an Analysis
It is a certainty that Tigraine was Rand's mother, Shaiel.
Tigraine was the Daughter-Heir of Andor, and was married to Taringail Damodred. They had a son, Galad. Tigraine and her brother Luc were sent to Tar Valon, in the usual tradition of the royalty of Andor. Tigraine vanished mysteriously from Tar Valon, never to be heard from again. [TEOTW: 34, The Last Village, 441-2]
In [TSR: 34, He Who Comes With the Dawn, 392-3], we learn about Rand's mother, Shaiel. Her tale corresponds marvelously with Tigraine's.
Add to this the fact that many people comment on how Rand looks like the Royal Family of Andor (Lord Barthanes [TGH: 32, Dangerous Words, 392], and many Andoran nobles [LOC: 26, Connecting Lines, 380]), the description of Luc [TSR: 33, A New Weave in the Pattern, 368]), and there you have it.
In LOC at one point in Caemlyn, Rand finally learns about Tigraine's story and is very upset until he figures out that he is not actually directly related to Elayne. Thus, he has placed himself in the family tree (see section 2.5.1) even if no one else has.
Yes, this means that Galad is Rand's half-brother.