I'm a California attorney, but it would be from your testimony, and the physical evidence. Please be sure to photograph the scene, any skid marks of either car, photo and video going both directions of travel of you and the other car, all damages to your car and the other car. Canvas the scene at the same time/day for any witnesses that may have been in the neighborhood, neighbors, store clerks, people sitting at bus stops, etc.
You would prove the liability of the other party through your testimony, under oath. Also perhaps photographs of your vehicle would show the nature of the impact, which would dovetail with your description of how the accident occurred.
Your situation does not differ from any accident situation where there are no witnesses other than the 2 drivers involved.
Mr. Lundeen is licensed to practice law in Florida and Vermont. The response herein is not legal advice and does not create an attorney/client relationship. The response is in the form of legal education and is intended to provide general information about the matter within the question. Oftentimes the question does not include significant and important facts and timelines that, if known, could significantly change the reply and make it unsuitable. Mr. Lundeen strongly advises the questioner to confer with an attorney in your state in order to insure proper advice is received.
The best way to establish liability is to leave it to professionals. You should report the crash to your insurance carrier. Ask them to send out an adjuster to look at the damage to your vehicle. There is a better than average chance that the adjuster can establish how the crash may have happened from the damage alone. Also file the instructions of the other attorneys who have commented here.
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