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勾股定理的一个有趣证明
下图中，ABC是直角三角形；C是直角。如图所示是勾股定理的一个证明。您看明白了吗？
耐人寻味的是这里用到了直角三角形的内切圆，与三边（两直角边和一斜边）分别相切于三点 D，E，F。过这三个切点的半径把圆分成三部分。而每一部分都是对称的！由此首先：
c = (a-r) + (b-r) = a+b – 2r
其中 c 是斜边长，a, b 是两直角边长，r 是这内切圆的半径。
我们于是有
c2 = (a-r + b-r)2
为证勾股定理，余下的就是把上式的右边变化成 a2 + b2. 建立的方法是通过切割面积c2 利用面积相等把切割的小块与 a2 + b2 中的小块完全匹配。(记 O 是内切圆的中心. 注意 D，E，F 是三个切点。）
看一眼如下的推演：
c2 = (a+b-2r)2
c2 = (a+b)2 – 4r (a+b) + 4 r2
c2 = a2 + b2 + 2ab – 4r (a+b-r)
在上面式子中，划掉尾巴上的两项，就是勾股定理。所以我们只要证明：
2 ab = 4r (a+b-r)
即（ab）/2= r (a+b-r). 注意！直角三角形的面积是啥？是 (ab/2). 直角三角形ABC（被过三切点的半径）分成三块：四边形 OFAD，ODBE 还有OECF（正方形）：面积分别是 r (b-r), r(a-r), 还有 r2. 加起来，ABC 的总面积不就是 r(a+b-r) ？由此（ab) /2 = r (a+b-r).
之所以说内切圆的使用耐人寻味，是因为在本证明中只是利用切点去分割三角形；割成的三小块：四边形 OFAD，ODBE 还有OECF（正方形）面积都与 r 有关。是不是很有趣？
啊哈! 证明毕。
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Pairing up with a Perfect Match
Math starts from the simple and goes a long way.
In this post, we will start from “pairing up” like 1 + 4 = 2 + 3 = 5 but there is a long way to go so that we learn the sets, working on the sets, pairing-up, observing and developing conditions for a perfect pair-up.
Pairing up for a Perfect Match
The formula on our face page of “amazing numbers” is rather interesting:
1 – (1 ⁄ 28) = (1 ⁄ 2) + (1 ⁄ 4) + (1 ⁄ 7) + (1 ⁄ 14)
The point of interest is that: if you look at all divisors of 28: they are 1,2,4,7,14,28; with the exception of 28 which is itself, all divisors have appeared in this formula, and they appear in the form of so-called “unit fraction”, where numerator is 1. So (1 ⁄ 2), (1 ⁄ 4), etc. are all unit fractions.
Indeed, we present a fraction equation to make it a bit unusual, but there is a low-pitch but straightforward ways to present number 28. We have that:
28 = 1 + 2 + 4 + 7 + 14
To get to the earlier fraction form, just divide every term by the number 28.
The smallest complete number is 6 (=1+2+3), 28 is the 2nd complete number, and after that, you will not see a complete number until 496. So complete numbers are rare among all positive whole numbers.
Complete numbers 6 also has a nice fraction form, as:
1 – (1⁄6) = (1⁄2) + (1⁄3)
This announces the opening of new CTCS math blog site.
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