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Brace yourselves, I'm gonna talk about sports.
If you slightly stretch your definitions, and don't listen to the lies big NFL is spreading, the Indianapolis Colts are actually the second oldest NFL team.
The Dayton Triangles (incredible name) were founded in 1913, starting off in the old Ohio League, and joined the NFL as a founding member in 1920. The Triangles are believed to have played in the very first NFL game in October 1920.
The Triangles then moved to Brooklyn and became the Dodgers. Side tangent: NFL teams really liked to copy MLB team names from the 20s through to the 40s, the New York Giants are the last big vestige of this, and the Pittsburgh Steelers were originally the Pittsburgh Pirates. There was an NFL and MLB St. Louis Cardinals at the same time for a couple decades, but it was a complete coincidence! The baseball and football Cardinals both date back to the late 19th century, but the football Cardinals started off in Chicago, coincidentally moving to St. Louis in the 60s, and now playing in Arizona. Early NFL teams also liked to pick names that thematically fit with the local baseball team, such as the Chicago Bears and Cubs, and the Detroit Lions and Tigers.
Anyways, back to the Triangles lineage. The Dodgers eventually renamed themselves the Tigers for a single season, and merged with the Boston Yanks for 1945 because of the war.
Here's where the headache begins, the Yanks and the Tigers separated again after the 1945 season, and the Tigers owner left for the rival AAFC to become the New York Yankees. In '49, the Yanks owner moved the team from Boston to New York to become the Bulldogs. When the AAFC merged with the NFL in 1950, only three teams were selected to join fully. The rest had their players dispersed to teams in their area. The Yankees players were dispersed between the Giants and Bulldogs, who renamed themselves the Yanks again.
In 1952, the Yanks sold the team to the league after financial problems. The league sold the team, which relocated to become the Dallas Texans (no connection to the AFL's Dallas Texans, who are now the Kansas City Chiefs)
The Texans sucked. They didn't even last the full 1952 season before being sold back to the league. So again, the team was sold to a new owner and relocated, this time to Baltimore. The Baltimore Colts (no connection to the AAFC's Baltimore Colts, one of the three teams chosen to fully join the NFL, which quickly folded after a year or two) were officially founded in 1953, and continue to this day in Indianapolis. A move infamously executed in the middle of the night under threat of imminent domain by the state, using Mayflower moving trucks. (Yes, the move was shitty, but Baltimore also stole the original Browns from Cleveland so not like their fans can claim to be saints either.)
The NFL doesn't consider the Colts lineage to trace back to the Triangles, and neither do the Colts themselves. It's a fun curiosity that lives on in the mind of fans, and sometimes it pops up every now and then.