角度

烟 · 发表于 2005-1-11 17:20:05

已知角AOB等于40度，P是角内一点，角POA＝30度。M、N分别是OA和OB上的一点，要使▲PMN的周长最小，角MPN应是几度？

fzy · 发表于 2005-1-15 20:22:38

Yes, thanks. You know what I mean.

husonghu · 发表于 2005-1-14 21:19:18

"When the perimeter reaches the minimum, we must have 角NMO = 角AMP (not NMA) and 角MNO = 角BNP (not MNB)",
I guess, hehe.

呵呵 · 发表于 2005-1-12 22:46:38

Sorry, 再加20度

呵呵 · 发表于 2005-1-12 20:55:18

80度

fzy · 发表于 2005-1-12 18:28:04

We assume P is a fixed point and we adjust M and N to achieve the minimum. Otherwise they all shrink to O.

When the perimeter reaches the minimum, we must have 角NMO = 角NMA and 角MNO = 角MNB. Because 角NMO + 角MNO = 180° - 角AOB, we have 角NMP + 角MNP = 360° - 2 * (角NMO + 角MNO) = 2 * 角AOB. Therefore 角MPN = 180° - 2 * 角AOB. In this particular case, it is 100°

This is a very nice problem with a very nice result.

BTW Mr or Ms 烟, I would like to post this problem to another forum if I have your permission. I will, of course, credit it to you and this forum.

cn · 发表于 2005-1-11 18:16:54

0度

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回复：Good answer, but there is a typo.....

Good answer, but there is a typo.....

Sorry, 再加20度

80度

100°

0度