Equivalent statement for affine subspaces

Question

We take $A$ to be a nonempty subspace of standard Euclidean space, then we have the following statements equivalent:

1. A is an affine subspace
2. $$\forall x,y\in A, t\in\mathbb{R}, (1-t)x+ty\in A$$

It looks like just following the definition but here is how to prove it only uses that affine set is a translated linear subspace. How can we show the two statements are the same?

Answer 1

Pick some $p\in A$ and define $A_p=A-p$. Then

1. Prove that $A_p$ is a vector subspace
2. Use that $A=p+A_p$ so it is an affine subspace

How to do part 1: Let $(x-p)\in A_p$ with $x\in A$. Then for $\lambda \in\mathbb R$ we have $$ \lambda(x-p)=\underbrace{((1-\lambda)p +\lambda x)}_{\in A}-p\in A_p. $$ Now its left to show that sums are contained