No, the third car owner has to pay both cars damages!

The third one's insurance has to pay the damages to both cars in front.  The ideal is you have police report.  Let Geico deal with them.  If Geico recovers the cost from the third car's insurance, you will get your deductable back.  Otherwise, you just lost your deductable.  Don't waste your time on dealing with professional insurance people when not necessary.