Veach began racing at age 12 when he was "discovered" by current IndyCar Series team owner Sarah Fisher's dad, Dave Fisher. In 19 months Veach had gone from go-karts to the cockpit of an open-wheel Formula BMW machine. Near the end of 2009, while testing a Formula BMW, Atlantic Championship team owner Eric Jensen signed Veach to his team for the 2010 Atlantic Championship season. In early March 2010, the Series halted operations.
U.S. F2000 National Championship
In his first year as a driver in the U.S. F2000 National Championship for Andretti Autosport, Veach had 10 top-five finishes and four podium results. Despite entering the series after the season-opener (St. Pete) and therefore missing out on two rounds of race points, he managed to finish fifth in the driver championship point standings. His performances also contributed to Andretti Autosport securing the team championship for the 2010 season. He was also named a semi-finalist for Sports Illustrated's annual Sports Kid of the Year award.
On December 9, Andretti Autosport announced they had re-signed Veach to again compete in the USF2000 National Championship and WinterFest in 2011.
In January 2011, Veach won the 2011 U.S. F2000 Winterfest - a winter race series held in Florida. Veach won two of the five races and won the title by six points over Andretti teammate Spencer Pigot. Veach won the 2011 U.S. F2000 National Championship season opener at Sebring International Raceway. Veach finished fourth in series points.
Star Mazda Championship
Veach made his Star Mazda Championship debut for Andretti Autosport at Infineon Raceway in August 2011. He also competed in the series' season finale at Mazda Raceway Laguna Seca, where he finished third. He competed in the full season for Andretti Autosport in the #77 car in 2012, finishing tenth in points with a best finish of third at Lucas Oil Raceway at Indianapolis.
Veach moved up to Indy Lights with Andretti Autosport in 2013. Veach led the most laps and finished on the podium in third at the Milwaukee Mile and won the pole at Auto Club Speedway on his way to seventh in points. He remained in the series and Andretti Autosport in 2014. He collected three wins and nine podiums in 14 races to end third in the standings behind Gabby Chaves and Jack Harvey. After being out of the cockpit for 2015, Veach will return to Indy Lights in 2016 with Belardi Auto Racing.
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