Complaint Review: Police Department

We were following our friend's car to their house in Fort Sheridan. Speed limit was 25 miles. Both cars, our friend's and our car were going at 23-24 miles. The police officer let our friend's car go, but stopped us. He told us we were going at 40 miles.

He asked as if we were following another car and when we confirmed that we did, he said that we must have been speeding trying to catch up. We told the police officer that we were not trying to catch up and that we were not speeding, but he insisted on his version of the story and gave us a ticket.

I never had speeding tickets before as I do not speed and I do not deserve this ticket now. My wife was in the car with me, she is my witness, my friend in the car we were following is my witness too. When I called lawyer to consult about this situation she told me that the police officer's word will be considered over mine and he did not show me the radar reading because he did not have to.
It looks like the police officer can say what ever he wants, but I am assumed to be guilty?!

The lawyer recommended to go to court and bring the witnesses with me. According to the lawyer I will have to pay her fee and any other charges the judge might decide on and this ticket might go on my record regardless.
My Allstate motor club pays only $50 for the lawyer, so I will be responcible for the rest of the atterney fees, any court fees plus it still might go on my record?!!!! Please help!!!!

I need help to resolve this issue and some convinsing not to loose my confidence in the police that is there to protect us, not set us up for something that we did not do.

Joe
Glenview, Illinois