Who was Keturah?
Rashi and Targum Yonasan: Keturah was alias Hagar.
Ramban (in Pasuk 66) and Rashbam (according to the simple P?shat): Keturah was another woman - not Hagar. 1
Why does the Torah call Hagar ?Keturah??
Rashi#1: Because her deeds were beautiful like the Ketores. 1
Rashi #2: Because she 'tied up' 2 her womb (she refused to marry any other man) from the time that she left Avraham.
Targum Yonasan: Because she was initially 'bound' to him from day #1.
In Vayeira, 21:14 Rashi commented that Hagar returned to the idolatry of her father's house? Refer to 25:1:1.1:1.
'Katar' is the Aramaic term for 'Kashar' (tied).
If Keturah was not Hagar, why did Avraham not go to Charan to find a wife, like he did for Yitzchak?
Ramban (in 25:6): Avraham was fussy only with regard to Yitzchak, who was destined to build K?lal Yisrael, but with regard to himself and his other children. 1
Refer to 25:6:2:1. In fact, the Ramban concludes, Keturah was from Cana'an. Otherwise, the Torah would have stated her nationality, as it did with regard to the wives of Eisav (see Bereishis 26:34). Refer to 25:1:2.1:1.
What is the significance of the fact that the Torah mentions Avraham marrying Keturah immediately after Yitzchak married Rivkah?
Bava Kama 92b: This is the source for the saying 'The tooth that is not eating suffers sixty pains when it hears its fellow-tooth eating. 1 Consequently, when Avraham saw his son married he followed suite.
Torah Temimah (citing the Aruch): Which means that a person who hears his friend eating and he himself is not eating, suffers 60 pains.
Why does it say "va'Yosef," especially according to Rashi, that he remarried Hagar?
Malbim: The primary seed of Avraham is Yitzchak. His other children are a mere secondary addition.
Ha'amek Davar: Avraham remarried in order to have more children 1 - "la'Sheves Yetzarah." This is even if their lineage does not follow him [e.g. if they are considered slaves], like it says in Yevamos 62a. 2
Ha'amek Davar: Alternatively, he remarried because "it is not good for man to be alone" (2:18).
The Gemara says that one even fulfills the Mitzvah of Peru u'Revu with such children. According to the opinion in Bava Basra 16b that Avraham did not have a daughter with Sarah, we can say that he remarried to fulfill Peru u'Revu!
QUESTIONS ON RASHI
Rashi writes: "She was called Keturah because her deeds were as pleasing as incense (Ketores)." But earlier, Rashi wrote that Hagar had reverted to her native idolatry (Rashi to 21:14)?
Gur Aryeh and Moshav Zekenim: She immediately did Teshuvah, upon experiencing Hash-m's kindness, when He sent her an angel and a spring of water (Bereishis 21: 17-19). Also see Keli Yakar.
Rashi writes: "Keturah - This was Hagar." Why interpret this way?
Gur Aryeh: If Keturah had not been introduced to us before, the Torah should state her origin - and not leave us to assume that she was a local. 1 Avraham had warned Yitzchak not to marry into Kena'an, and certainly he himself would not do so! Rather, it was Hagar (who we know was Egyptian - 16:1).
