bodybuilding and cycling

Hello,
I'm new to this forum so am not yet totally familiar with the layout.
I've been into bodybuilding for several years and train on the usual split system. I'm not huge by any means but I have bench-pressed around 120 kilos on the Olympic bar and have done my share of very intense full-squats.
I used to suffer bad knee-pain and crippling arthritic pain in my hips as I seem prone to joint stiffness. However, I started cycling and found my hip-pain disappeared. I also found it worth-while to do sufficient cycling to improve my fitness.
I bought a second-hand road-bike for only 200 pounds and enjoy my cycle runs to the gym (which is located in the hills).
The general view is that you can't realy mix cycling and heavy weight-training since one sport is anaerobic while the other is aerobically based.
I wonder, though, if any people have tried to mix a strength sport with an activity such as cycling. Is there a happy balance that can be found and could my years of squats even be a plus where hill-climbing is concerned? By the way, I'm heavy for a cyclist (weighing close to 200 pounds).