Sudden brake engagement

why does my toyota camry sudenly engage brakes at 60 mph

Model year?
Engine type?
Transmission type?
Under warranty?
Odometer mileage?
Maintenance and repair history?

That seems unlikely. Why do you think the brakes are being engaged as opposed to some engine or transmission problem?

2000 AUTOMATIC CAMRY
200,000 MILES

Well, clearly it is not under warranty, and I guess that you are unsure of the car’s maintenance history.
An uncertain maintenance history is always a concern, but I digress for the moment.

If the car is actually engaging its brakes without any input from you (and, as Tardis indicated, this is unlikely), then I would suggest that you have your mechanic change the brake fluid and inspect the flexible brake lines. It is possible for there to be deterioration inside these lines and for this to interfere with normal functioning of the brakes, particularly if the brake fluid was not changed every 30k or so. However, a visual inspection of the outside of the flex lines is unlikely to reveal anything about the inside of those lines. Still, this is something to consider.

As Tardis suggests, what feels like brake engagement is more likely to be either a driveability (engine) problem or a transmission problem, and both of those issues are almost always tied to lack of maintenance.

So, what is the maintenance history of this car?

THIS IS MY FATHER’S CAR AND HE SAID THAT HE HAS HAD ALL THE SCHEDULED MAINTAINANCE DONE ON SCHEDULE. HE SAID THAT THE WHOLE CAR VIBRATES SEVERELY AT JUST OVER 60 MPH AND USE TO STOP WHEN HE TAPPED THE BRAKE PEDAL. BUT NOW HE HAS TO BRING THE CAR TO A COMPLETE STOP AND START OVER FOR IT TO STOP VIBRATING. HE IS FINE AS LONG AS HE STAYS UNDER 60 MPH.

Okay, so then the brakes are not engaging by themselves, and instead, the car is exhibiting a severe vibration problem at high speeds. There are lots of possibilities, but the first thing to look at is the front end, i.e.–steering and suspension components.

At 200k miles, it is very likely that ball joints and tie rod ends are worn out. This is a definite safety issue, so I would suggest that he get the car to a qualified mechanic a.s.a.p., and to avoid driving it at high speed. If he is very lucky, it will just turn out to be imbalanced tires, but all the same this needs to be fixed right away.

Oh, and please don’t type in “all caps”. That is the internet equivalent of shouting, and is considered to be impolite.