Anime of TYPE-MOON work, except Fate/stay night, are:

Shingetsusan Tsukihime: is Anime about a boy named Shiki who had car accident and he pretty much messed up his mind. When he woke up he saw lines on anything, even human. When he touches those lines it hurts him a little, but if he able to cut them they fell apart. He gained mystic eyes of death perception. Then the Magican gave him glasses that make those lines disappear. And he lived his life peacefully, until he got into rage and killed woman who was in park where he got berserk... the rest is up to u to find out.

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Kara no Kyokai...

Saber true identity is Arturia Pendragon (King Arthur)... note: when I refer to he I mean King Arthur and when I say she I mean to Saber (Arturia)

King Arthur was gathering all his knights to look for the Holy Grail from which Jesus drank. The legend berkata he could gain with it eternal life... So he searched for it his whole life. But in the end he fought with Mordred. U can see in Anime gap from that fight when Saber was fighting herself... In the end anda could see as she was in the end when she was laying wounded sejak the tree. Deeply wonded she started to regret for even drawing sword from the...

About less then 50% of artikel is recap from anime, my opinion of Rin's thoughts, but it has strong point in connection so read it all...

Connection of Archer and Rin:
Why Rin summoned Archer- Heroic spirit Emiya?
In order to summon a servant anda need catalyst (something that he possessed in past, something precious).
Shiro was wounded sejak Lancer and was near death, hopefully Rin made it in time to heal his would with her precious jewel that was passed down in her family in generations. After she was done healing him she regrouped with Archer and they went home. She left that used up jewel, with...