Fringed Milkwort

We found these tiny wildflowers in several places along the Lake Huron shore, usually in the shade of pine trees. I thought at first they were orchids, but after looking at my photos decided they could not be. The leaves were not right for an orchid and they did not fit the description of the plant I had assumed them to be. A friend in Michigan identified them for me as Fringed Milkwort.