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[几何] 直角三角形斜边折线长度的取值范围

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hbghlyj posted 2019-5-12 01:53 |Read mode
Last edited by hbghlyj 2019-5-13 22:19在Rt△ABC的斜边BC上取点P、Q,使得BP=CQ,AC上取点R,AB上取点S
(1)求四边形PSRQ周长的取值范围,
(2)求折线PSRQ的取值范围
QQ浏览器截图20180909213316.png

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realnumber posted 2019-5-13 15:04
A是直角顶点,对吗?

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kuing posted 2019-5-14 02:12
回复 3# hbghlyj

手机上的我也不会插……这论坛没有手机版,本身就不太适合用手机看……

最小值作对称:
QQ截图20190514021033.png
所以最小值就是斜边(两问都是);

至于最大值,如果不允许四边形的边自交的话,则:
(1)当 P、Q 取 B、C 并且 R、S 的任一个取 A 时,四边形的周长就是 `\triangle ABC` 的周长,除此之外的一般情形会有线段位于三角形内部,周长就会小于它,所以最大值就是 `\triangle ABC` 的周长;
(2)母鸡,有空再想。
但如果允许四边形的边自交(按照题目的叙述,应该是可以的),那会复杂一点,还是有空再想,或者弃坑……

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original poster hbghlyj posted 2021-6-16 05:26
回复 4# kuing
手机端已经可以插入图片了,我把原来的3#删了哈

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original poster hbghlyj posted 2021-6-16 05:30
Reply kuing
Let′s settle the remaining problem (2)

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