Letzanim

Rashi at the beginning of the parsha says the Torah repeats Avraham begot Yitzchak since שהיו ליצני הדור אומרים מאבימלך נתעברה שרה וכו'. מה עשה הקב"ה? צר קלסתר פניו של יצחק דומה לאברהם, והעידו הכל - אברהם הוליד את יצחק, וזהו שכתב וכו' שהרי עדות יש שאברהם הוליד את יצחק.  Why are they called ליצנים presumably they are reshaim for saying lashon harah?  Furthermore, if they didn't believe someone so old could give birth they should have claimed Sarah couldn't give birth for Avraham already showed he could have a son, for he had Yishmael?  The Beis Halevi says that they were not denying who the parents were, there claim was why did Sarah only give birth now, at such a late stage of life, it must be the fact that she was taken by Avimelech and she had to endure that pain was a merit for her to have a child, וזהו שאמרו בדרך ליצנות מאבימלך נתעברה שרה וכמו הליצנות שאומרים פלוני בנה בית מחמאה, והכוונה שע"י שעשה סחורה בחמאה והרויח ממון בנה הבית, וע"כ קראם ליצני הדור.  This was the ליצנות.  However, since in time the point would be lost and people would take it literally that Yitzchak is a son of Avimelech, ויהיה לעז חלילה וע"כ צר קלסתר פניו כשל אברהם.  

His grandson, the Briskor Rav, takes a different lesson from the Rashi:

In the Mir parsha sheet, Rav Chayim Shmuelevitz is quoted as having said as to why the letzanim would claim Avraham is out of the picture and not Sarah, that one can not a question on a letz.  Presumably, he means that if one wants to scoff at something they will do so even if a non-sensible manner.  One can't debate a letz with logic. 

Rav Ezrachi asks why does a miracle have to occur just to debunk the claim of letzanim?  He explains that the point isn't just to prove the letzanim wrong.  Rather, Avraham's mission in life is to get people to come closer to Hashem.  If they don't believe in the continuity of Avraham, they are in denial of his mission and that will bother Avraham.  It is the fact that there are letzim unconvinced by Avraham that demands a miracle in order to make Avraham happy. 

The Rebbe asks why would Yitzchak not look like Avraham in the first place if he is his biological son?  He explains the middot of a person are reflected in their countenance and since Avraham is chesed and Yitzchak is din they would have looked different (Likutay Sichos volume 20, 30.)