I'm looking for an English name of "定比分點公式". There's a page on the Chinese Wikipedia dedicated for the vector version of this formula, but that's only available in Chinese. I copy the relevant section of its content so that you don't have to click the linked Wiki page to see what it's about.
$${\overrightarrow {AD}}={\frac {|{\overrightarrow {CD}}|}{|{\overrightarrow {BC}}|}}{\overrightarrow {AB}}+{\frac {|{\overrightarrow {BD}}|}{|{\overrightarrow {BC}}|}}{\overrightarrow {AC}}$$
Here's the first case of the area version of this formula that is well-known in the Chinese literature. It can be found in many Chinese journals and books, say Zhang Jing-zhong's (張景中) book 新概念幾何 (, which literally means "new concept geometry" and which has not been translated into other languages). I take the following picture from Fan's Math World.
$P$ and $Q$ are on the same side of the plane divided by line $AB$, and line segment $PQ$ doesn't not intersect with line $AB$. Point $C$ lies on line segment $PQ$ such that $PC = \lambda PQ$. Then we have $$S_{\triangle ABC} = \lambda S_{\triangle QAB} + (1 - \lambda) S_{\triangle PAB}.$$
Remarks:
- $S_{\triangle ABC}$ means "the area of $\triangle ABC$".
- I didn't put the word "segment" in between "line $AB$". To understand why, consider the second case of the area version of this formula: line segment $PQ$ intersect with line $AB$ at point $M$. Then $\triangle MAB$ degenerates into line segment $AB$ or $AM$ or $BM$, so it has area zero. However, on the right hand side, the area of both $\triangle PAB$ and $\triangle QAB$ and both coefficients $\lambda$ and $1-\lambda$ are positive. Hence, we have zero LHS and positive RHS, which is absurd.
P / | A----B---M \ | Q
Both versions make use of the a point on a line segment whose distance with the endpoints follows a fixed ratio. I asked on Discord Math server, and a guy told me that he would call it something like "convex interpolation", but I didn't find relevant pages on DuckDuckGo the search engine.