Grace Millane, 22, was last seen in central Auckland on December 1 and was reported missing on Wednesday.
At a press conference on Saturday afternoon, Detective Inspector Scott Beard said police had located a 26-year-old man at a central Auckland address who was helping police with their inquiries.
Beard said officers had gathered evidence on Saturday that had “established this is a homicide”.
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He said Grace Millane’s family had been told and were “devastated”.
“We are treating Grace’s disappearance as a homicide investigation.”
“We still do not know where Grace is. We are determined to find her and return her to her family,” he said.
“We know that he [the man police are talking to] was with her on the Saturday evening and went to a number of places before going to the CityLife hotel.”
Police had identified and found a vehicle of interest and officers were trying to find out where it had been, he said. He would not confirm what type of vehicle it was.
“I know that the thoughts of all New Zealanders will be with Grace’s family tonight and I want to reassure everyone both here and abroad, that NZ Police are determined to do everything we can to find Grace,” Beard added.
Earlier on Saturday, police said some of Grace Millane’s belongings, including her passport, watch and a necklace, were missing from the hostel she was staying at.
The news comes hours after her family said they were “staying very positive” as the search for her continued.
In a statement released on Saturday morning (New Zealand time), they said it had been “an extremely difficult and emotional time for us all”.
Her father David Millane fronted a press conference at Auckland Central Police Station on Friday after flying to New Zealand from his home town of Essex in the United Kingdom.
A tearful Millane appealed to the public to come forward with any information, “no matter how small”.
“I would just like to take this opportunity to appeal to anybody who has seen, spoken to, or come into contact with Grace over the last few days, to come forward with any detail, no matter how small, and contact the investigation team,” he said.
The last confirmed sighting of Grace Millane was at 9.41pm Saturday at the CityLife Hotel in Queen St, when she was seen with a “male companion”.
She was wearing a black dress and white shoes.
The man, who lives in New Zealand, was the same man being questioned by officers at Auckland Central Police Station on Saturday afternoon.
Officers have conducted a scene examination at an apartment at the CityLife Hotel and were trawling through “hundreds” of hours of CCTV footage as part of the investigation.
Dozens of calls had been made to the 0800 investigation helpline number, and that information continued to be reviewed and assessed, Beard said.
Much of the contact was from members of the public offering accommodation and assistance to the Millane family.
In the statement issued on Saturday, Millane’s family praised Auckland police “who have been outstanding in their support and efforts in helping to find Grace”.
Grace Millane was on a year-long OE after graduating from the University of Lincoln in September.
The advertising graduate and keen artist arrived in New Zealand on November 20, after travelling through South America.
Photos on her Instagram showed she travelled to Cape Reinga with friends before returning to Auckland.
This was her first time travelling abroad alone.
Described as “fun loving, outgoing and [a] family-orientated person”, Millane was never out of contact with family and friends for this long, her family said.
Anyone with information that could help with the investigation has been asked to contact 0800 676 255.