Shabbos, Chapter 9, Mishnah 3. This mishnah continues to discuss the scriptural sources for various halachos, presenting three halachos together with their sources in the verses.
The law of a woman who discharges semen:
The first halachah deals with a woman who discharges semen after having been with her husband. The rule is that one who touches semen becomes a first level of tumah and must immerse in a mikvah. The woman too must immerse after relations, but if she afterward discharges semen - even after she has already immersed in a mikvah - she becomes tamei once again, since she has come into contact with the semen.
This is only the case, however, as long as the semen is still viable. The question our mishnah addresses is at what point the semen loses its viability. The baraisa in the Gemara records various opinions on this matter, and our mishnah adopts one of them and seeks its source in the verses.
"Minayin l'poletes shichvas zera bayom hashlishi shehi temei'ah?" - From where do we know that a woman who discharges semen on the third day after relations is considered tamei, meaning that the semen is still present and capable of producing a child?
For it was stated at the giving of the Torah, when the husbands and wives were commanded to separate from one another and not to have relations, so that the man and the woman would be tahor at the time they received the Torah. The purpose of the separation was that if the woman would afterward discharge semen, the period during which the semen is viable would already have passed, and even if she discharged it after immersing in a mikvah it would not render her tamei.
The wording of the verse: "Heyu nechonim lishloshes yamim" - prepare yourselves for three days. And the verse continues (although the mishnah does not quote it): "Do not approach a woman," that there be no marital relations. From here we learn that throughout all three days the semen retains its vitality, and even on the third day its vitality remains and the woman becomes tamei when she discharges it - and for this reason they were required to wait three days.
Washing the circumcision on the third day that falls on Shabbos:
The second halachah deals with an infant after his bris milah. The infant is regarded as being in a state of possible danger to life, and therefore one may desecrate Shabbos on his behalf and wash him in hot water after the circumcision. The mishnah says: "Minayin shemarchitzin es hamilah bayom hashlishi shechal lihyos b'Shabbos?" - From where do we know that it is permitted to wash the child on the third day after his circumcision when it falls on Shabbos?
For it is stated regarding Shimon and Levi, after they caused the men of Shechem to circumcise themselves: "Vayehi bayom hashlishi bihyosam ko'avim" - that they attacked them on the third day while they were still in their pain. From here we learn that the third day is a time of great pain. According to some opinions this is derived by a kal vachomer, that it is certainly permitted to desecrate Shabbos on the earlier days as well; and some say that the law was stated specifically regarding the third day.
A crimson thread on the head of the goat sent away:
The third halachah: "Minayin shekoshrin lashon shel zehoris b'rosh sa'ir hamishtale'ach?" - From where do we know that a red thread is tied to the head of the goat that is sent out to the wilderness on Yom Kippur to be killed?
For it is stated in Yeshayah: "Im yihyu chata'eichem kashanim kasheleg yalbinu" - that if your sins are red like the red color of the crimson dye, they will turn white as snow. From here we learn that a red thread must be tied, for the red thread turns white once the sins have been forgiven.
In summary: This mishnah presented three halachos together with their sources in the verses: the tumah of a woman who discharges semen on the third day, derived from "Heyu nechonim lishloshes yamim"; the permission to wash the circumcision on the third day when it falls on Shabbos, derived from "Vayehi bayom hashlishi bihyosam ko'avim"; and the tying of a crimson thread on the head of the goat that is sent away, derived from "Im yihyu chata'eichem kashanim kasheleg yalbinu".