今天看到了一道很有趣的几何题。如图，四边形 ABCD 中，连接对角线 AC 、 BD ，若 ∠ABD = 40° ， ∠ADB = 80° ， ∠CBD = 70° ， ∠CDB = 50° ，求 ∠BAC 的度数。
这道题看上去似乎非常简单，但稍作尝试你就会发现，仅仅是在这几个角度之间来回倒腾，是没法求出 ∠BAC 的度数的。听说过 Langley 问题（就是那个臭名昭著的 20-80-80 三角形）的人就会知道，这种类型的题目往往会非常非常地复杂。据说这是 1989 – 1990 年加拿大亚伯达省中学数学竞赛中的一道题目，当时只有一个人做对，并且解答过程用到了非常繁琐的三角函数运算。然而，这道题实际上有一个非常漂亮的秒杀方法，完全不需要使用三角函数。你能想到吗？
答案：把注意力集中在三角形 ABD 上。容易看出， ∠A = 180° – 40° – 80° = 60° 。另外， CB 平分 ∠B 的外角， CD 平分 ∠D 的外角，因而 C 就是三角形 ABD 的一个旁心。这说明， CA 也将平分 ∠A 。因此， ∠BAC = 60° / 2 = 30° 。
参考资料： Andy Liu, “A Better Angle From Outside”, Mathematical Horizons, November 1997
太妙了！
真漂亮！
CD 既平分 ∠B 的外角，又平分 ∠D 的外角<–这里写错了。。。
回复：谢谢提醒，已改正
真心漂亮啊
应该是：另外，CB平分∠B的外角，CD平分∠D 的外角，因而…
回复：谢谢提醒，已改正
看前留名~
#_# 厉害。。
看完题目之后想的是: 难道有比旁心还简单的做法?…
延长AB到E使得BE=BD，连接CE，DE
角EBC=70度所以三角形BDC和三角形BEC全等，所以角BCE=60度
于是角ECD=120度，角BAD=60度，所以ADCE四点共圆
所以角问号=角EDC
而三角形EDC是一个顶角为120的等腰三角形
所以角问号=角EDC=30度
有点复杂不过也做出来了。。。
我和楼上英雄所见完全相同啊。。。
这次我做出来了~和标准答案一样~话说第一次看到这题目，同学和LS的想法一样要做圆。（然后他对着近似椭圆的圆画了一条鱼。。。）
这题小花有介绍，象这类球角度问题有连几何方法都特别复杂的。。数学吧的亡牌口香糖是这方面的专家。
我也做出了，过ADB外接圆O1，过BCD做外接圆O2,AC交圆O1于点E。弦BD在圆O2所对的圆周角DBC为60度,角DEB为120度，可知E为圆O2的圆心。连DE和BE。可知DE和BE为圆O2的半径，故DE=BE.所以角DAE=角EAB.所以，角EAB=30度
这让我想起了90年代国家队某年的集训题中的一个角度问题，也是相当有趣。这类问题三角函数是比较容易做的。
我也是文科生，已经四年没碰过数学了，我刚做了下题，后看了你的答案，我已经忘了旁脚这个概念了，我用的方法是在过d做直线交ab于e,使三角形ade为直角三角形，同理做直线bf交cd于f，使三角形cbf为直角三角形，即证明cd平行ab,在根据其他条件，可得出答案是30度。
不好意思，我弄错了
真的是要注意观察呐
四个小角设四个未知数，列4个方程，暴力解之。
秒解。。。
我的水平就高中水平，然后，乃的辅助线添加之后直接用外角和内角的知识求就够啦～
我对旁心这概念就没什么印象，数死早了。几乎是完全没印象啊（不敢说死，但就是没印象），不知道广大高中理科生有多少人知道或熟悉这个概念？
我只能想到坐标法……这题有意思
大神跪拜
不知道列方程能不能算出来~？？
直接做平行线，就可证明了，为什么搞那么复杂呢？
我2了，平行线不行。
△ABC中，sin110/sin?=AC/BC (1)
△BCD中，sin50/sin70=BC/DC (2)
△ACD中，sin(60-?)/sin130=DC/AC (3)
三个式子相乘，左边sin110与sin70消去，sin50与sin130消去，右边是1，
所以sin(60-?)/sin?=1,?=30。
只是解析几何没有欧氏几何证得“漂亮”。
以BCD为内角，BC为边作正三角形BCM，由CDA+MBA=180得四点共圆，得MAB=50=MBA，所以A是BCM外心。这似乎很好想
A不是△BCM的外心。。
猜测等于20度似乎也说的通
初中党表示刚学旁心。。。
人之所以平凡，在于无法超越自己。
竞赛党表示208080是基本模型。。。那个旁心这道题也做了五六遍了。。。
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