They were found dead at their home in nearby North Royalton on Sunday.
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North Royalton police identified the victims, who were Taylor Pifer, 21, Kylie Pifer, 18, and their mother Suzanne Taylor, 45.
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On Tuesday, Stark County Sheriff George Maier said that Brinkman had been employed by the couple for several years. The couple's bodies were found Monday in their North Canton home.
He had apparently dated the family's daughter, authorities said.
Stark County filed murder charges against Brinkman on Tuesday.
A look at attacks in Britain in recent weeks and years
Armed police guard a street in the Finsbury Park area of north London where a vehicle hit pedestrians on 19 June 2017. She said her thoughts were with the injured, their loved ones and emergency officials who responded to the incident.
Lt. Robert Safran told Cleveland-based WEWS the man in the house is connected to "the incident in North Royalton, (Ohio)" on Sunday.
Brinkman was arrested early Tuesday after a nine-hour standoff on Valley Forge Drive in Brunswick.
He was jailed from October 18, 1999 to March 2, 2000. When he stopped by the house, he would later tell a 911 operator that he saw "her friend George's car" at the home.
Meanwhile, this morning the Stark County Sheriff's Office said Brinkman is a suspect in the murders of a couple found dead Monday at their home in Lake Township, which is about 45 miles south of North Royalton.
A responding police officer said the three slain women were cold to the touch, "face down in bed all lying right next to one another", according to a police report obtained by the station.
Cleveland19 reported Monday, citing police, that the mother and daughter were still alive Saturday afternoon and were killed sometime that evening. The bouquet was still outside the home Sunday. It is not clear if he has yet been charged in the three other deaths, in North Royalton, or if he has retained a lawyer or entered a plea.
Brinkman is scheduled to appear Thursday morning in Parma Municipal Court, North Royalton Detective Dave Loeding said.
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