A quasigroup \((Q,\cdot)\) is said to be medial if \((x\cdot y)\cdot(z\cdot t)=(x\cdot z)\cdot(y\cdot t)\) for all \(x,y,z,t\in Q\), and is called idempotent if \(x\cdot x=x\) for all \(x\in Q\). If \((R,+,\cdot)\) is the coordinatizating ring of a translation plane and the kernel of \(R\) contains at least one element \(k\) distinct from 0 and 1, then a groupoid \((R\times R,\cdot)\) may be defined via \[ (x_ 1,x_ 2)\cdot(y_ 1,y_ 2)=(k\cdot x_ 1+y_ 1-k\cdot y_ 1,k\cdot x_ 2+y_ 2-k\cdot y_ 2). \] The author shows this groupoid is an idempotent medial quasigroup with the following property: (*) If \(A\) is any line and \(d\) is any element of \(R\times R\), then \(A\cdot d=\{a\cdot d; a\in A\}\) is a line and \((A,\cdot)\) is a subquasigroup of \((R\times R,\cdot)\). It is also shown that if in any affine plane a quasigroup \((Q,\cdot)\) can be defined such that \((Q,\cdot)\) satisfies (*), then the affine plane is a translation plane whose coordinatizing ring possesses a nontrivial kernel. Using the above results the author then generalizes some work of \textit{J. Šiftar} [J. Geom. 20, 1-7 (1983; Zbl 0514.51010)] and \textit{N. K. Pukharev} [Mat. Issled. 71, 77-85 (1983; Zbl 0551.20052)] on affine planes over left distributive quasigroups.