The function to be minimized is the square of the distance \(\displaystyle f(x, y, z)= x^2+ y^2+ z^2\) subject to the constraints \(\displaystyle g(x, y, z)= x+ y+ z= 1\) and \(\displaystyle h(x, y, z)= x^2+ y^2= 1\).

The first thing I would notice is that, since \(\displaystyle x^2+ y^2= 1\), the function to be minimized reduces to \(\displaystyle f(z)= 1+ z^2\). That is minimized when z= 0. And the condition that x+ y+ z= 1 means that x+ y= 1 so y= 1- x. Using, again, \(\displaystyle x^2+ y^2= 1\), we have \(\displaystyle x^2+ 1- 2x+ x^2= 2x^2- 2x+ 1= 1\). So \(\displaystyle 2x^2- 2x= 2x(x- 1)= 0\). \(\displaystyle x= 0\), in which case \(\displaystyle y= 1\) or \(\displaystyle x= 1\), in which case \(\displaystyle y= 0\).

The solutions are \(\displaystyle (0, 1, 0)\) or \(\displaystyle (1, 0, 0)\).