First, let's look at the differences between the female and the male
Lucanus placidus, first 2 from the left. The female mandibles are as if you were joining the index fingers and your thumbs. The male mandibles, centre, are as if you were joining your 3 middle fingers, note the tiny thumb below.
Second, let's look at the male
Lucanus capreolus, the last one on the right, note how curved and pointed its mandibles are by comparison, one knob only.
After having worked all this out by myself I found among my papers that the best key to these species is in
A Manual of Common Beetles of Eastern North America, I happen to have a copy of the section with the Family Lucanidae, courtesy of Charles Staines.
Many thanks to my readers, site visitors rather, in particular Rae Ann Gallati for sending me the
Lucanus placidus specimens, luckily a boy and a girl :-)