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Kate was last seen by her dad on Wednesday, May 30 at around 6.30pm.
It is believed she went to meet a date at the CC Club in Piccadilly Circus.
The sales manager then called a friend at 8.20pm saying she was in Chiswick and texted another friend at 8.50pm saying she was in Richmond.
Kate's car was last spotted on Chiswick Bridge at around 11.58pm on that day - and it is thought she may have visited the Chiswick, Richmond and Twickenham areas.
Officers, who had been searching for her car, say they found the vehicle, and another they had been looking for in Harlesden.
"We believe Kate and the man may have called at the Roebuck pub during Wednesday evening, and may have crossed the road to the terrace and park area opposite.
"We would welcome any information from members of the public who may have seen any activity out of the ordinary, or who have seen Kate or a man fitting the description, or her grey VW Golf that evening.
"We are also very keen to hear from anyone who may have seen the couple later that evening in or around Oxhey Woods."
Police today confirmed they were combing Petersham Meadows on Richmond Hill for clues in connection with the case.
Pals had set up a group titled Help Find Kate on social networking site Facebook. The group currently has more than 1,000 members.