I just proved, using lengthy and awkward trigonometric calculations, the following theorem:
Let $\mathcal{E}$ be an ellipse with foci $E$ and $F$, let $P$ be a point outside the ellipse, and let $PT_1$, $PT_2$ be the two tangents from the point $P$ to the ellipse $\mathcal{E}$, where $T_1$ and $T_2$ are the points of tangency. (It does not matter which of the two contact points is $T_1$ and which is $T_2$.) Then $\measuredangle{T_1PE} = \measuredangle{FPT_2}$ (this is an equality of oriented angles).
This "property of the tangents to an ellipse" is already mentioned in this post, where the author of the post says that he recently stumbled upon it, but does not tell whether he proved it himself or came upon it in a textbook or a paper or, perhaps, on some website. He gives the following reference to the property, but there is no proof there, and no hint of its provenance. So I am wondering whether this theorem is already long known, and in case it is, to whom it is attributed. It might be one of Poncelet's, or it may go as far back as Apollonius of Perga.
Yes, it's already long known. You can see a short proof of this theorem in George Salmon, A treatise on Conic Sections, art. 189, page 166: https://archive.org/details/treatiseoncon2ed00salmuoft/page/166
The book was published in 1850.