What is the English name of the result known as the "Shared-Edge Theorem" in Chinese?

Question

There is this theorem that I think is pretty useful, but I don't know the English name. It's called 共边定理 in Chinese which literally means Shared-Edge Theorem. There is a Chinese Wikipedia article about it at https://zh.wikipedia.org/wiki/%E5%85%B1%E8%BE%B9%E5%AE%9A%E7%90%86.

_{(source: hudong.com)}

The theorem says that following for this triangle:

$$\frac{\triangle PAB}{\triangle QAB}=\frac{PM}{QM}$$ I need to know the English name for this so I can reference it on my tests. As you can see, this works because a side is shared.

Answer 1

To expand upon my comment, here's an excerpt from Grünbaum and Shephard's 1995 article "Ceva, Menelaus, and the Area Principle" (JSTOR link) in Mathematics Magazine.

The purpose of this note is to show that [the theorems of Ceva, Menelaus, and Hoehn] and other results, and their extensions to general polygons with arbitrarily many sides, are the consequences of a simple idea which we shall call the area principle. [Description of equality between ratios of segments and associated areas, essentially matching OP's description of the Shared-Edge Theorem.]

Emphasis mine. Grünbaum and Shephard appear to have coined this name for the result. I'm not sure how widely-known the name is.

The article is quite interesting, by the way. The authors have always had a real knack for generalizing things in unexpected ---and unexpectedly-broad--- ways.

Answer 2

Also called the "area method" in addition to "area principle" -- to the extent that such minor distinctions matter -- according to Shephard's 1999 "Euler's Triangle Theorem" that cites the 1995 Grünbaum & Shephard paper mentioned above. 1