An officer does not have to witness a crime to charge someone with it. Even a ticket. I recommend that you hire a ticket attorney. Their fees are reasonable and they will always save your money.
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If you decide to have a trial on the matter, it can be harder for an officer to properly answer questions on cross exam if he didn't witness the incident. However, it's not impossible--in NY, an example might be an officer who testifies that skid marks were present after a rear end accident on the highway may be able to successfully sustain a charge of following too close.
Yes it is legal. Whether the testimony of the officer will hold up in court is another story . You need an attorney to desk with this as a guilty verdict can have consequences beyond just the ticket. There will be a civil action with respect to the collision .
Yes, the officer can issue a ticket after the fact. When the report was filed, apparently you told the officer what happened, your side of the story. You told him that you were making a u-turn after a left. That is an admission. It will be used to prove that you were making a u-turn. Whether or not you were making it unsafely is what is at issue. Just because you were making a u-turn and a collision occurred doen't mean you caused it. You need a lawyer.