TITLE: What is the name of this theorem of Jakob Steiner's, and why is it true?
QUESTION [46 upvotes]: In The Secrets of Triangles a remarkable theorem is attributed to Jakob Steiner. 
Each side of a triangle is cut into two segments by an altitude. Build squares on each of those segments, and the alternating squares sum to each other.

The book doesn't include a proof, and I'm not sure how to start.
Does this theorem have a name? How could one go about proving this beautiful relationship?

REPLY [5 votes]: (Community wiki.) As pertains to the claim that this was proved by Steiner, Viktor Blåsjö‏ points to Steiner's Werke, volume I, page 189 in a tweet accompanied by the following two images:


For completeness, here is the figure with the corresponding identity in $*$ illustrated geometrically:

where the summed area for the red and blue squares, respectively, are equal.

The above generalization, where the squares aren't determined specifically by altitudes (ie, by perpendiculars from the orthocenter) but rather by perpendiculars from an arbitrary point $P$, is known as Carnot's Theorem, not to be confused with Carnot's Theorem.  Could this be one of those cases where the "wrong" name has been attached to a result? (Note that Steiner (1796–1863) and Carnot (1753–1823) were contemporaries.)