locus of midpoints in space (Oral Moscow Team MO 1999.2.BS4)

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High School Math Grades 9-12, Ages 13-18

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locus of midpoints in space (Oral Moscow Team MO 1999.2.BS4)

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#1 h Nov 11, 2020, 7:05 AM

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Two crossing perpendicular lines are given in space. Find the set of midpoints of all segments of a given length whose ends lie on these lines.

This post has been edited 1 time. Last edited by parmenides51, Nov 11, 2020, 7:06 AM

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#2 h Nov 11, 2020, 7:09 AM

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ok im just gonna say it: stop ~~spamming~~ putting multiple posts just for one question which clogs up the forum and pushes all other problems by others out of the way. I recommend you put it all into one post. especially since you're just taking it all from one source

This post has been edited 1 time. Last edited by LightningGirl, Nov 11, 2020, 7:15 AM

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#3 h Nov 11, 2020, 7:19 AM

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each problem set in contest collections comes from one source, one contest each year, but by posting it separately one may see many different solutions
these problems are collected inside aops here

This post has been edited 2 times. Last edited by parmenides51, Nov 11, 2020, 7:19 AM

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#4 h Nov 11, 2020, 7:21 AM

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This is a pretty common occurrence in physics problem (like a ladder falling down a wall)

Let's say the length of the segment is $a$ and the 2 perpendicular lines are the cartesian coordinate axes.
Then we can parametrize the location of one end point as $(\pm a\cos t,0)$ and the other endpoint as $(0,\pm a\sin t)$ .
The midpoint is then $(\pm \frac{a}{2}\cos t,\pm \frac{a}{2}\sin t)$ , which is a $\boxed{circle}$ with radius $\frac{a}{2}$

@2above I personally find it nice that parmenides51 has taken the time to post these problems since HSM isn't a super active forum. Since people have the ability to bump posts, and parmenides51 is introducing new problems to the community while also providing a service to the contest collection, I don't see what's wrong with this.

This post has been edited 3 times. Last edited by natmath, Nov 11, 2020, 7:24 AM

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#5 h Nov 11, 2020, 8:37 AM

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Me too. Thanks @paramenides of taking the initiative for posting these in HSM.

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