Relation of length of a projection of a point to a line

Question

In the given figure, can it be said that $x \leq a + b - d$?

Answer 1

Due to Pythagoras, $d = \sqrt{a^2-x^2} + \sqrt{b^2-x^2}$. However, $a - \sqrt{a^2-x^2} $ goes faster to zero than $x$ for $x\to 0$: $$ \lim_{x\to0}\frac{a-\sqrt{a^2-x^2}}{x}=0,$$ so your inequality does not hold for small $x$.

Answer 2

The answer is No. The inequality $x\le a+b-d$ does not hold for $(a,b,d)=(9,2,10)$ because $$x=\frac{\sqrt{(a+b+d)(a+b-d)(a+d-b)(b+d-a)}}{2d}=\frac{3\sqrt{119}}{20}\gt \frac{3\sqrt{49}}{20}= \frac{21}{20}\gt 1.$$