Kaczmarz's and Cimmino's projection methods

For a consistent system $A_{n\times r}x=b$ with $\operatorname {rk}\left(A\right)=r$, scale the rows so that $\left\| A_{i*}\right\|_{2}=1$ for each $i$, and let $H_{i} = \left\{ x \mid A_{i*}x=b_{i}\right\}$ be the hyperplane defined by the $i$th equation. Beginning with an arbitrary vector $p_{0}$, the Kaczmarz sequence is generated by the double loop For $k=0,1,2,3,\ldots$ For $i=1,2,\ldots ,n$ $p_{kn+i} = p_{kn+i-1} - (A_{i*}p_{kn+i-1}-b_{i})(A_{i*})^{T}$ Prove that the Kaczmarz sequence converges to the solution $x$ of $Ax=b$ by showing

In 1938 the Italian mathematician Gianfranco Cimmino used the following elementary observation to construct an iterative algorithm for solving linear systems. For a $2\times 2$ system $Ax=b$, let $H_{1}$ and $H_{2}$ be the two lines (hyperplanes) defined by the two equations. For an arbitrary guess $r_{0}$, let $r_{1}$ be the reflection of $r_{0}$ about the line $H_{1}$, and let $r_{2}$ be the reflection of $r_{0}$ about the line $H_{2}$. As illustrated below, the three points $r_{0}$, $r_{1}$, and $r_{2}$ lie on a circle whose center is $H_{1}\cap H_{2}$ (the solution of the system).

The reflections r_1, r_2 of an arbitrary point r_0 about lines \mathcal{H}_1, \mathcal{H}_2 lie, together with r_0, on a circle centered at \mathcal{H}_1 \cap \mathcal{H}_2.

The mean value $m = (r_{1}+r_{2})/2$ is strictly inside the circle, so $m$ is a better approximation to the solution than $r_{0}$. It's visually evident that iteration produces a sequence that converges to the solution of $Ax=b$. Prove this in general by using the following blueprint.