Yup, those are the ends of some of my tall spiral gingers. After the bloom finishes, they start to make a bulblet near the top, so I cut below the bulblet and put them in water. Roots form just below the bulblet.
I'll bet your begonias have done so well this summer because of all that rain you had. Mine are doing better now that it's been raining a bit more here. July? Nuh uh, they were not happy! I'm thinking partly it's that my begonias aren't great with my high pH well water when I have to irrigate them. I have one circuit of sprinklers that runs off the cisterns, and when the begonias are over there, they do better, too. But it's sort of out of the way for me, and the snails get to them over there . . . can't win!
Pam, take some cuttings off those long, leggy begonias! This time of year they will root really easily, just stick 'em in a pot of potting soil in the shade and don't let them dry out. Here are some of mine, first one is a new start this spring of one Mj gave me, Maurice Amy. Second are cuttings of B. odorata started in May.
Looking Glass, Don Miller, huge plant of Corallina de Lucerna with some Rex begonias to the left.
This one is my namesake, B. Elaine and has nearly black leaves and raised dots.