The Squared Norm of a Convex Combination of Two Points

Let $n$ be a natural number with $1\le n$, let $\mathbb{R}$ be the real numbers with the order $\le$ of their ordered field structure, and for a real number $s$ write $s^{2}$ for the power $s\cdot s$. Regard Euclidean space $\mathbb{R}^{n}$ as a real vector space, with the sum of points and the scalar multiple, and write $x-y$ for the difference of points. Let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on $\mathbb{R}^{n}$.

Then for all $x,y\in\mathbb{R}^{n}$ and every $t\in\mathbb{R}$,

$$\lVert t\,x+(1-t)\,y\rVert^{2}=t\,\lVert x\rVert^{2}+(1-t)\,\lVert y\rVert^{2}-t(1-t)\,\lVert x-y\rVert^{2}.$$