Medicine and sickness heal each other.  If there was no sickness, nothing would be considered medicine.  If there were no medicine, sickness could not be cured and would just be considered the way things were, like old age, death, and taxes. 

Many medicines (antivenom, antibiotics) are made from the very poisons they seek to cure (medicine arising from sickness).  Too much medicine leads to drug dependency and/or chemical toxicity (sickness arising from medicine).

These are interesting observations (at least to me) but lead us no closer to solving the koan.  How can they not be the same thing and not not the same thing?

To answer the question, find a place to sit still for a while.   Pay close attention to the breath and let go of your thoughts - don't try to stop them, but don't engage in them either.  Be still.  Be quiet.  As your thoughts dissolve into the background, you might find yourself in a realm beyond thought and judgement, beyond naming and identifying things.  When conscious thought disappears - ta-da! - there's the answer to your koan.