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Now answer this and I'll pay you a bounty.
Suppose the trees are circular with radius . Answer the same two questions as above.
Can I answer :P
Sure, you will steal @Undisciplined's bounty but you snooze you lose
For any diameter greater than 0, it's impossible for a line of sight not to fall into a tree. Since any irrational number can be approximated by a rational number, for any diameter greater than 0, there will be a finite rational approximation whose error falls within that margin. So, in a more formal way: for any tree diameter greater than 0, all lines of sight fall into a tree, thus there's 100% of probability that any line chosen at random will fall into a tree in a finite distance.
Excellent answer. Well stolen.
Lol! Thank you. Here, half the bounty, you prompted it.
That's very generous. All I did is spout off my first intuition.
good job. another way to say this is that there does not exist any continuous interval of real numbers $(a,b)$ with $b>a$ that doesn't contain a rational number within it.
Good one, yes. I think the way I expressed it is more easy to visualize, what do you think?
Yours is more intuitive. If you've already setup whatever it is you need to about irrational slopes, this is probably more of a slam dunk.
Thank you! And yes, indeed, hence my temptation to steal it :P
Thank you!! :)