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Can it be shown mathematically that the line connecting tips of crescent is parallel to the North-South line and the line gives latitude of the plane ( When the Moon is sufficiently close to the horizon, we can approximate the surface of sky as a simple plane.)

Sources :

https://olympiads.hbcse.tifr.res.in/wp-content/uploads/ino20/INAO2020-Solutions-20200204.pdf

https://www.naturalnavigator.com/find-your-way-using/moon/

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They don't. In this image (from 28th Dec 2021) the blue grid represents lines on an equatorial grid. The lines that run from top right to bottom left are North-South lines. The tips of the crescent very clearly don't point to the North.

enter image description here

The moon is lit by the sun and both the sun and the moon are in the plane of the ecliptic (to within a few degrees). So, as the "bulge" of the crescent points towards the sun, and along the ecliptic, The tips of the crescent will point towards the ecliptic pole which is about 23 degrees away from North.

However since the moon is not exactly on the ecliptic even this is not exactly correct, and close to new moon the tips of the moon could point almost anywhere (think how the invisibly thin cresent will rotate as the moon passes the sun on a new moon at which there is no eclipse.)

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Actually, you can do this with the setting first quarter moon, which is a special case. The following was noted in the answer given to a similar question, recently -

The angle between the lunar line of demarcation and horizon at, or near, the time the first-quarter moon sets, can be easily determined on any date when the moon is in its first quarter, as the sum of the solar declination on that date and latitude of the observer.

The answer is in regard to a similar aspect given here. However, a line connecting the tips of the crescent will only be parallel to the north-south line at two specific times of the year.

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