A geometry problem by Toni Shani

We know that a geometric progression is a sequence in which the ratio of a term to its preceding term is a constant number $r$ .Now in the right triangle we know that $\triangle ABC\sim \triangle AHB$ from this we have $\frac { AB }{ BC } =\frac { BH }{ AB } =r$

So we have that $BC,BA,BH$ is a geometric progression and the first sentence is correct.

Now in the second sentence we know that $\triangle AHC\sim \triangle BHA\Longrightarrow \frac { AH }{ BH } =\frac { CH }{ AH } =d$

So $BH,AH,CH$ is a geometric progression and the second sentence is wrong