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brainpushups

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$$\frac{2x}{x+z}$$

and

$$\frac{4xy}{(x+y)+(y+z)}$$

Where, for the sake of this thread let's say that ##x<y<z## But really all that Huygens requires is that y is an intermediate value; x does not need to be the smallest value.

What is to be shown is that the second expression is always larger than the first. Furthermore, it is at its largest when the value of y is the geometric mean of x and z. I chose to take the ratio of the expressions to get

$$\frac{(x+y)(y+z)}{2y(x+z)}$$

I guess I don't have much experience with this sort of thing; I'm not even sure how to go about the first part which perhaps seems rather trivial (or maybe not). Just to be clear - I'd like to show that the fraction above is always greater than 1 for ##x<y<z## and that this ratio is greatest when ##y=\sqrt{xz}## Can anybody offer either some insight or a suggestion of what to further study that would help me gain some experience for how to go about this? Huygens does it by using line segments to compare quantities and ultimately comparing areas formed by rectangles of the segments but, as I said, it was challenging to follow in detail.