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You might. I might not.
I do enough (too much) time on the bike. Gym was refreshing and fun for me. It didn't seem to slow me down last year. My coach is injured though so I'm looking for a new one.
Have a read:
https://www.trainingpeaks.com/blog/year-round-strength-training-for-cyclists/
https://www.theguardian.com/lifeandstyle/2009/jan/11/cycling-strength-training
http://www.cyclingweekly.co.uk/fitness/strength-training-for-cyclists-is-it-worth-it-125222
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I am a very weak and puny man with legs like twigs. I write enjoy 'strength' work in the gym for the legs an core conditioning for variety and to add a little power and robustness.
I do sometime question why i'm doing it though...
It can occasionally feel detrimental. The DOMS if i've not done a session in a while or if i've upped the weights can fuck me the next day, but if you get a regular routine its ok. If i had less training time i would ditch the gym though.
Am I alone in thinking that for us amateurs, you'd see more gains by spending more time doing well-structured on-bike training than spending time getting to/from the gym and doing a few reps? Gyms are also horrid places full of disgusting people.