僅使用圖中給定的角度以及初等幾何原理來解角x的度數(shù)，并給出詳細(xì)步驟；

若需要輔助線，僅使用沒有刻度且寬度忽略不計(jì)但長(zhǎng)度不限的直尺與圓規(guī)。

(圖下面的句子意思是：圖片并非是按比例畫的，即不允許使用量角器)允許使用的初等幾何原理：

對(duì)頂角相等；

鄰補(bǔ)角互補(bǔ)；

兩直線相交，有且只有一個(gè)交點(diǎn)；

兩條平行線間的距離相等；

三角形內(nèi)角和為180°；

兩條平行線被第三條直線所截，同位角、內(nèi)錯(cuò)角相等，同旁內(nèi)角互補(bǔ)(其逆亦真)；

等腰三角形兩腰相等，兩底角相等，頂角的角平分線、底邊上的高和中線重合(其逆亦真)；

等邊三角形三邊相等，三角相等且均為60°(其逆亦真)，每一角的角平分線、其對(duì)邊上的高和中線重合；

直角三角形有一個(gè)90°的角(其逆亦真)；

相似三角形對(duì)應(yīng)角相等(其逆亦真)；

全等三角形的四個(gè)判定定理如下(其逆亦真)：

邊角邊：有兩邊和它的夾角對(duì)應(yīng)相等的三角形全等

邊邊邊：三邊對(duì)應(yīng)相等的三角形全等

角邊角：兩個(gè)角和兩角夾的邊對(duì)應(yīng)相等的三角形全等

角角邊：兩個(gè)角和其中一個(gè)角所對(duì)的邊對(duì)應(yīng)相等的三角形全等；

在直角三角形中，30°角所對(duì)的直角邊等于斜邊的一半。

全等三角形的斜邊、直角邊判定定理不可用，同樣不能用正弦余弦定理和三角函數(shù)知識(shí)。

以下是英文版的關(guān)于解題過程的提示。

Proofs may be written informally using plain English. Just be sure to include all the steps in your reasoning, or at least all the key steps.Providing a diagram is very helpful but not required. You can draw a diagram on the computer or you can draw it on paper and then scan it or photograph it with a digital camera.

Please number your steps. This makes it easy for both writer and reader to talk about the steps.

Name each point you use with a letter (ex., say "point A" or simply "A"). Identify lines with two letters (ex., say "line AB" or simply "AB"). Identify triangles with three letters(ex., say "triangle ABC" or "tri ABC" or "△ABC" or simply "ABC"). Identify angles with three letters, vertex in the middle (ex., say "angle ABC" or "angABC" or "∠ABC" or simply "ABC").

If you don't provide a diagram, you will need to describe the named points with words (ex., say "the intersection of AE and DB is G").Even if you provide a diagram, you must define with words each new line that you draw, in order (ex., say "Draw a line through C perpendicular to AB intersecting AB at H").

Justify all key steps (ex., say "AC=BC because ABC is isosceles"). You may omit the justification when simply "chasing angles" — calculating angles based on any of these simple rules: triangle angles sum to 180, ?supplementary angles sum to 180, opposite angles are equal.

以下是簡(jiǎn)單翻譯(translated by 甜甜)

使用平淡的英語來寫證明過程可能更樸實(shí)無華一些。只要保證你的證明過程包括了所有的證明步驟，至少是關(guān)鍵步驟就可以了。可能一個(gè)圖解很有用，但是這并不必要。你可以在電腦或者紙上畫一個(gè)圖，然后用數(shù)碼相機(jī)拍下來。

請(qǐng)給你的步驟標(biāo)號(hào)。這使步驟對(duì)于作者或讀者來說變得更好討論。(如何翻譯這句話使我糾結(jié)了十分鐘左右——甜甜注)

把你使用的每一個(gè)點(diǎn)都用字母標(biāo)上號(hào)(比如說“點(diǎn)A”或者只用“A”).使用兩個(gè)字母來確定一條線(比如說“線AB”或者只要“AB”).使用三個(gè)字母來確定一個(gè)三角形(比如“三角形ABC”或“三角ABC”或“△ABC”或者只要“ABC”).使用三個(gè)字母來確定一個(gè)角，其頂點(diǎn)在中間(比如說“角度ABC”或“角ABC”或“∠ABC”或只要“ABC”).

假如你不給出一個(gè)圖，你需要給出新的點(diǎn)的描述(比如，“令A(yù)E與DB交于G”(或者AE與DB的交點(diǎn)為G，因?yàn)椤傲預(yù)E與DB交于G”可以被理解為延長(zhǎng)AE交DB于G——甜甜注))縱然你給出了一個(gè)圖，你也必須合理描述你作出的每一條輔助線(比如，過C作AB的垂線交AB于H)。

證明所有步驟(比如“AC=BC因?yàn)?三角形——甜甜注)ABC是等腰三角形”)。當(dāng)你僅僅在“追求角度”(這個(gè)詞沒有什么合理的漢語對(duì)應(yīng)詞，即我們平時(shí)說的來回“倒”角——甜甜注)時(shí)你可以省略不必要的步驟——在以下規(guī)則中之一為基計(jì)算時(shí)：三角形內(nèi)角和為180°，補(bǔ)角之和為180°，對(duì)頂角相等。

以下是作者自述(為了與上邊區(qū)分開，這句話采用了藍(lán)色字體)

Sorry, but I'm not giving the answer nor the proof here. You will just have to work on it until you either solve it or are driven insane. If you email me at, I may give you a bigger hint (if I feel like it). If you think you have ?solved it, you can ask me if your answer is correct, but please also tell me how you got the answer. The proof may be written informally, but you need to tell me all the steps, or at least the key steps, in your solution. It is helpful if you also send me a diagram. Try to persuade me that you are not just guessing. I have additional small, medium, and large hints, but you must first show your efforts to convince me that you have struggled valiantly.

Please don't search the the web for the answer _That's cheating.You will only deprive yourself of many hours of delicious frustration.Of the proofs posted on other websites, some are valid proofs and some are not.

I did not invent these problems. After I first read problem 1, I worked on it for many hours over several days before I eventually figured it out.A couple of years later I came back to the problem, but I had forgotten my proof. It took me many hours to figure it out again! Problem 2 also took me many hours to solve.

How hard are these problems? Any teenage student can understand the proof,but very very few are able to discover the proof on their own. Of the people who have emailed me (more than a thousand), I'd estimate only one or two percent (mostly math professionals and college students) have provided valid proofs without significant hints. (The hints given above are not significant hints.)

Based on my emails, most people who think they have found a proof are wrong. Most do not even have the correct answer for angle x. Of those that have the correct answer, the proofs they send me are usually wrong (incorrect or incomplete). People who say "it only took me a few minutes" almost never have a valid proof.

These problems have been published in many places. Problem 2 first appeared here: Langley, "A Problem", Mathematical Gazette, 1922.http://www.drgarygruber.com/geniuschallenge.htm says his high school teacher showed him problem 1 in about ? ? ?1955. Tom Rike says problem 1 first appeared in print here: Harry Schor,The New York State Mathematics Teachers' Journal, 1974. It also appeared ? ? ?here: "Problem 134", Eureka (now Crux Mathematicorum), 1976. ? ? ?Dr. Gruber popularized problem 1 in several papers (such as "The Genius Test") which appeared in newspapers throughout the 1990s (Universal Press Syndicate and Los Angeles Times Syndicate). That's where I discovered it.

Notice that I call this the "world's hardest easy geometry problem", not the "world's hardest geometry problem". The world's hardest geometry problem would be something really hard, like the Poincaré conjecture.

以下是簡(jiǎn)單翻譯(translated by甜甜)

抱歉，我既沒有給出答案也沒有給出過程。在你解決掉或者瘋掉之前(還真好意思說呵——甜甜注)你必須一直奮發(fā)圖強(qiáng)地研究它。如果你給我往發(fā)電子郵件(為了尊重原作者，沒有去掉外鏈——甜甜注)，我高興的話我就給你一個(gè)提示。如果你認(rèn)為你解決了它，你可以發(fā)郵件來問我對(duì)不對(duì)，但是請(qǐng)告訴我你是怎么解決的。證明可能很平淡，但是你需要告訴我所有的步驟，或至少是所有關(guān)鍵的步驟。如果你可以給我發(fā)一個(gè)圖，那就更好了。你得讓我知道你不是瞎猜的。我還會(huì)追加小提示、中提示和大提示，但是你必須讓我看到你的努力，讓我相信你確實(shí)"英勇地拼搏"過。

請(qǐng)不要上網(wǎng)找答案——那是欺騙。這只會(huì)使你經(jīng)受幾個(gè)小時(shí)的有趣的挫折。網(wǎng)上的證明，有一些是有效的，但是一些根本就沒用。

我不是這兩個(gè)題的發(fā)明者。當(dāng)我讀完第一問后，我做了幾天零幾個(gè)小時(shí)才終究華麗地做出來。兩年之后我又回來看這題，但是我忘了我的證明。我又花了幾個(gè)小時(shí)才再做出來一次！第二問也讓我花了幾個(gè)小時(shí)去證明。

這些問題到底有多難呢？所有青少年學(xué)生都可以看懂這個(gè)證明，但是非常非常少的人可以自己證明出來。在一千多個(gè)給我發(fā)電子郵件的人中，我估計(jì)只有1%到2%的人(大多數(shù)是專家或者大學(xué)生)不需要很大的提示就給出了有效的證明。(上面的幾個(gè)提示都是無關(guān)緊要的。)

從我的電子郵件上看，大多數(shù)認(rèn)為自己給出了證明的人都是錯(cuò)的。有些人連角x的度數(shù)都算不對(duì)。在給出了正確答案的人中，他們寄給我的證明大多是錯(cuò)的(證明有誤或未完成).那些說“我只用了幾分鐘就做出來了”的人幾乎沒有一個(gè)給出有效證明的。

這些問題已經(jīng)在很多領(lǐng)域出版。第二問首次于1922年在蘭利，“一問”數(shù)學(xué)公報(bào)上出現(xiàn)。格魯伯·加里博士說他的高中老師大概在1955年給他看過第一問。湯姆·里克說第一問首次于1974年在《紐約州數(shù)學(xué)老師日?qǐng)?bào)》中出現(xiàn)。它也于1976年出現(xiàn)在尤里卡的“問題134”.格魯伯博士在二十世紀(jì)九十年代的一些報(bào)紙(比如格魯斯競(jìng)賽)上普及過第一問。我也是在那里發(fā)現(xiàn)的這個(gè)問題。(翻譯這段花了我半個(gè)小時(shí)時(shí)間查資料——甜甜注)

注意：我稱它為“世界上最難的簡(jiǎn)單幾何體”，不是“世界上最難的幾何體”。世界上最難的幾何體可能真的很難。比如龐加萊猜想。

以下是問題一的微型提示。

A working diagram that is large and accurate will help you find the solution. Draw the diagram yourself using a protractor or print this correctly scaled diagram.

圖片：Triangle1Big.gif

以下是簡(jiǎn)單翻譯。(translated by 甜甜)

畫一個(gè)大的、準(zhǔn)確的圖能幫助你解題。自己用量角器或者打印一個(gè)按比例的準(zhǔn)確的圖。

(圖片下面的句子是：“此圖是按比例畫的”。意為可以使用量角器度量角x的度數(shù)——甜甜注)

以下是問題一的小提示。

To solve the problem, you will need to draw a few more lines inside the ? ?triangle.

You will need to do more than merely adding and subtracting angles.

The solution has more steps than you might expect.

以下是簡(jiǎn)單翻譯。(translated by 甜甜)

要解決這個(gè)問題，你需要在這個(gè)三角形中作幾條輔助線。

來回“倒”角是做不出來此題的。

做出此題所需步驟比你想象的多。

(問題二的微型提示同問題一——甜甜注)

以下是問題二的小提示。

Problem 2 is not solved in the same way as problem 1.

以下是簡(jiǎn)單翻譯。(translated by 甜甜)

問題2的方法和問題1不一樣。

總評(píng)：這文章的作者是不是閑的？為了一個(gè)數(shù)學(xué)題，我翻譯了一下午英語！

——甜甜

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