If three numbers are in H.P., then the numbers obtained by subtracting half of the middle number from each of them are in

Let the three numbers in H.P. be $a, b, c$. Then

$b=\frac{2ac}{a+c}$

Numbers obtained by subtracting $\frac{b}{2}$ (half of the middle number) are

$a-\frac{b}{2},b-\frac{b}{2},c-\frac{b}{2}$

or $a-\frac{ac}{a+c},\frac{ac}{a+c},c-\frac{ac}{a+c}\text{ or, }\frac{a^2}{a+c},\frac{ac}{a+c},\frac{c^2}{a+c}$

Clearly, these numbers are in G.P .