Despite what some people say it really doesn't matter how you cut the patch provided you cut it so that you can peel it off properly with each half. Just open the sealed pouch thingy and snip,. Then pop the unused half back into the pack, fold over the opening, put it back in the box or wherever you keep them and use the other half later.
There could be a tiny amount of deterioration along the cut edges but this would be negligible. I would prefer to cut so the patch is rectangular rather than triangular only because I feel the pointed edges may come unstuck more easily. Also this makes it easier to gradually increase the size/dose of the patch in a more controlled way. Just my view though. There will be others happily cutting theirs diagonally. Whatever works for you.
I always apply mine in the morning every third day but if after a shower then make sure you don't use oily/very moisturising shower cream etc so it sticks well. I think some women clean the area of application, again ie after showering to remove traces of product though I never do.
Good luck and let us know how you get on
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Hurdity x