AM and GM

$x$ is the geometric mean between $a$ and $b \implies x^2 = ab$

$y$ is the geometric mean between $b$ and $c \implies y^2 = bc$

$a,b,c$ are in A.P $\implies 2b = a + c$

A.M of $x^2, y^2 = \large\frac{x^2 + y^2}{2}$

$\implies \large\frac{ab + bc}{2} = \frac{b(a + c)}{2} = \frac{b \times 2b}{2} = b^2$

Let $d$ be the common difference for these terms. You get $a, a + d, a + 2d$ by the definition of AP.

$\implies x = \sqrt{(a)(a + d)} = \sqrt{a^2 + ad}$

$\implies y = \sqrt{(a + d)(a + 2d)} = \sqrt{a^2 + 2d^2 + 3ad}$

$\dfrac{x^2 + y^2}{2} = \dfrac{\underbrace{a^2 + ad}_{x^2} + \underbrace{a^2 + 2d^2 + 3ad}_{y^2}}{2} = \dfrac{2a^2 + 4ad + 2d^2}{2} = (a + d)^2 = \boxed{b^2}$