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These kind of Adventitious Angles problems are popular and there is a Japanese book https://www.amazon.co.jp/%E3%83%A9%E3%83%B3%E3%82%B0%E3%83%AC%E3%83%BC%E3%81%AE%E5%95%8F%E9%A1%8C%E3%81%AB%E3%83%88%E3%83%89%E3%83%A1%E3%82%92%E3%81%95%E3%81%99-%E2%80%954%E7%82%B9%E3%81%AE%E4%BD%9C%E3%82%8B%E5%B0%8F%E5%AE%87%E5%AE%99%E5%AE%8C%E5%85%A8%E3%82%AC%E3%82%A4%E3%83%89-%E6%96%89%E8%97%A4-%E6%B5%A9/dp/4768703402 gave quite complete solutions but it's via a very general approach and usually not feasible for middle school students.

You can read some analysis on this topic in Chinese at here: https://www.zhihu.com/topic/21763175/top-answers

https://zhuanlan.zhihu.com/p/105111160

These questions don't seem complicated, but if you think about it a little, you will find that they are not trivial. Plane geometry enthusiasts have given this type of problem a vivid name - "angular lattice point". The "angular grid point" problem can be forcibly calculated using the angle element Ceva's theorem, and will eventually boil down to some trigonometric functions identities. This method lacks beauty, and often involves guessing the answer first and then trying to figure it out, which feels like it doesnâ€™t touch the essence. After checking some information, I found that the "corner grid point" problem originated from a British mathematics teacher, Edward Mann Langley.

In general, there is no other sources that you get these Adventitious Angles problems because there is no math exams and homework problems chosen among them. You get the problems because (1) you are interested in solving math problems or geometric problems on internet from social media, (2) you did not walk away easily. and sometimes you are requested to write down how you solve them, what you think about them, what are your current conclusions and progresses, and yes, i have posted quite a lot of them on this site.

And my habit is to solve all this kind of problems in geometric approach whenever i see one from social media. So i have solved quite a lot of them and record them down in my notes.

FAQ: "Where are you stuck on this special one?"

Ans: The only reason i post this question here is because I have not seen a pure elementary geometric solution. If i have seen one after posting i will answer it myself. And that might due to the problem seems not so directly can be resolved by simple reflection or constructing equilateral triangles. Usually you need find a way to construct some equilateral triangle, or isosceles triangle so that the angle chasing can get to what could not be got otherwise. Sometimes circumcenter is needed by the theorem that "The central angle is twice an inscribed angle that subtends the same arc.", so if there is an isosceles triangle that the top angle is double of another angle formed by the same bottom vertices then the top vertex is the circumcenter of the triangle. This is quite often used trick in constructing solution for this kind of problems.

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