Why was the 1972 AFC Championship played in Pittsburgh?

Thank you for your question. Yes the game was played at Pittsburgh.
The Dolphins won 21-17 at Pittsburgh (Dec. 31, 1972, 50,845 people attended).
Back then, the playoff system wasn't as it is today.

In 1972 the NFC champ. game between Dallas and Washington wasn't played in Washington even though Washington won the NFC east ahead of Dallas and the 1973 game between Minnesota and Dallas was played at Dallas even though Dallas went 10-4 and the Vikings were 12-2 (the Vikings won anyway).

Ran across this question by accident over 6 years after the original post and thought I should clarify the answer as no further explanation was given at that time...

From 1970 through 1974, after the merger of the AFL and NFL, teams were not seeded by winning percentage. Home playoff games were determined by a preset rotation through each of the three divisions. Select division winners would play all playoff games at home, regardless of the comparative record of their opponent. The Wild Card team would play all playoff games on the road. In the 1972 regular season, the AFC Eastern Division Champion Miami Dolphins had gone undefeated yet played the AFC Championship Game in Pittsburgh because the AFC East was not in the "home game" rotation that year and Pittsburgh was. The Steelers were also not the Wild Card team. Starting in 1975, seeding for the playoffs was determined by winning percentage, then other tiebreakers if necessary.

http://www.profootballhof.com/history/story.aspx?story_id=1940

Also note that the 1972 NFC Championship was in fact played in Washington, as erroneously stated in the previous contribution. Dallas was the Wild Card team and, by rule, had to play every playoff game on the road.

http://www.pro-football-reference.com/boxscores/197212310was.htm

My reaction to this question is that it may be based on the issue of why an original NFL team, the Steelers, was playing in the AFC playoff- the conference where all of the AFL teams entered after the AFL-NFL merger of 1970. The answer to that is -- In 1970 The Pittsburgh Steelers move from the NFL Century Division to the AFC Central with the merger of the American Football League and the NFL.

The AFC was created after the NFL merged with the American Football League (AFL) in 1970.[1] All of the 10 former AFL teams along with the NFL's Cleveland Browns, Pittsburgh Steelers, and the then-Baltimore Colts joined the AFC. The two AFL divisions (East and West) were more or less intact, while the NFL's Century Division became the AFC Central.

https://en.wikipedia.org/wiki/American_Football_Conference