1994 Ford Probe Ford Probe won't turn over, but will run if

Hello, I have a 94 Probe, and i got in it the other day after driving it all day no problems. It wouldn't start the radio worked but wouldn't turn over at all. I tried push starting it to test my luck, and it started and i drove it all the way home 30 miles, the voltage was at normal so i dont think its the alternator. The starter doesnt make the clicking noise. I dont think its the battery because the radio was working with the key turned over. and also because it didnt even try to turn over. Is this a relay problem, or some kind of fuse problem. I checked the fuses and none seemed to be blown i could be wrong though. if you have any information or troubleshooting tips that would be appreciated... thanks...