The Torah Portion of Vayeira 5773
"...and he was sitting at the entrance of the tent in the heat of the day." (Bereisheet 18:1)
The Sages say that Avraham sits next to the opening of Gehinom and doesn't allow anyone to enter that is circumcized, and that is what is hinted at in the verse by "sitting at the entrance of the tent in the heat of the day". The "heat of the day" is referring to Gehinom which burns like fire.
"...and he ran toward them from the entrance of the tent..." (Bereisheet 18:2)
Why did he already start running from the entrance of the tent? The explanation is that according to the Halacha, someone who is going to do a Mitzvah needs to run, and someone who is returning from doing a Mitzvah needs to walk slowly. And if he returns from doing one Mitzvah and is going to do a different Mitzvah, if the two Mitzvot are of equal value, then the first half of the way he needs to walk slowly and afterwards he needs to hurry. But if the second Mitzvah is greater and more important, then he needs to hurry from the start. Therefore, since the Mitzvah of Hachnasat Orchim (hospitality -- receiving guests) is greater and more important than receiving the face of the Shechina (the Divine Presence), he needed to run already as soon as he started going to do the Mitzvah, and that was "from the entrance of the tent".
"...Shall I hide from Avraham that which I am doing...For I have known him, because he commands his children and his household after him" (Bereisheet 18:17-19)
Hashem revealed to Avraham that he wanted to overthrow Sodom, and we need to understand the connection between Hashem's need to notify Avraham about the destruction of Sodom and Hashem's knowledge that Avraham "commands his children". Those who explain the Torah say that this can be understood by means of a parable. When an old man enters a store to buy himself a garment, he tries on a lot of clothes until he finds a garment that fits him. In contrast, a father of a large family doesn't check the size of the clothes, but rather takes everything that looks suitable because in any event he won't lose out. If a garment doesn't fit one of his bigger children it will fit one of his smaller children. Similarly, when the Holy One Blessed Be He wanted to overthrow Sodom, he wanted Avraham to pray for them. And even though the prayers would not be effective for the men of Sodom, who were very evil, and those people would be removed from the world, in any event the prayers of Avraham Avinu would not be wasted. They would be a merit for his children and for the generations after him, and that is the explanation of "he commands his children and his household after him".
"And Hashem remembered Sarah..." (Bereisheet 21:1)
Rashi says that someone who prays for his friend, and he needs the same thing (for which he is praying on behalf of his friend), the one who is praying is answered first (i.e., his needs are fulfilled before his friend's needs are fulfilled). It is told that someone came to complain before a particular Tzaddik (Righteous Man), that his friend had opened a store next to his own store and was selling the same type of merchandise. The Tzaddik answered him, pray that customers will come to your friend and in any event, you too will be blessed because of your prayer.