Houston police said they were investigating the crash in the 1800 block of Westheimer Parkway, which is near Freedom Park.
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The person on the bike is a woman, HPD said.
Life Flight responded to the scene, police added.
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A day after the incident, police backtracked on their information that someone had died in the collision.
A 21-year-old woman was hospitalized in critical condition. The driver - a 69-year-old woman - showed no signs of intoxication, was questioned, and released, police said. The motorist wasn't immediately facing charges.
According to HPD, the SUV driver was following another vehicle in front when it made a sudden move to the left and a bang occurred. The driver realized she hit the cyclist in front of the other car.
Police added that they are not looking for the other driver.
Editor's note: Houston police originally called the Westheimer Parkway crash a fatality incident. Police, however, no longer referred to it as a collision with a death a day after it happened.