Thanks, gals! Although the silk is quite a bit heavier than, for instance, a silk blouse, I think you're probably right about stabilizing the whole triangle. I hadn't thought as far as the (eventual!) quilting. I might do a few test squares to see what works best.
The fabric came from my cousin, who purchased it on a recent trip to Japan. She's an artist and art professor who works primarily in "found" items - turning things that other people would normally throw away into art. I once attended one of her shows, and afterward sent her an e-mail stating that, yes, she's still an artist and I'm still an accountant. Hence her challenge to me to use my quilting creativity and her "found" objects in a project. I'm going to a retreat this weekend, and figure the endless repetition of sewing triangles together will allow me to both get something accomplished and yak with my sister and other retreaters.