每日彙整: 2017-08-07

樹莓派、樹莓派之學習、樹莓派之教育

GoPiGo 小汽車︰格點圖像算術《投影幾何》【四‧平面國】《辛》

若問如何區分人的『學過』與『學會』呢？這『九點圓』是個很好之『問題』︰

人們如何發掘『現象』間的『關係』？怎麼探討事物的『性質』 ？因何能發現且證明隱晦的『數學定理』？！也許讀讀維基百科上『九點圓 』詞條一小段文本︰

九點圓

九點圓（又稱歐拉圓、費爾巴哈圓），在平面幾何中，對任何三角形，九點圓通過三角形三邊的中點、三高的垂足、和頂點到垂心的三條線段的中點。九點圓定理指出對任何三角形，這九點必定共圓。而九點圓還具有以下性質：

  • 九點圓的半徑是外接圓的一半，且九點圓平分垂心與外接圓上的任一點的連線。
  • 圓心在歐拉線上，且在垂心到外心的線段的中點。
  • 九點圓和三角形的內切圓和旁切圓相切（費爾巴哈定理）。
  • 圓周上四點任取三點做三角形，四個三角形的九點圓圓心共圓（庫利奇－大上定理）。

200px-Triangle.NinePointCircle.svg    九點圓

EulerCircle4

Even if the orthocenter and circumcenter fall outside of the triangle, the construction still works.

歷史

1765年，萊昂哈德·歐拉證明：「垂心三角形和垂足三角形有共同的外接圓（六點圓）。」許多人誤以為九點圓是由而歐拉發現所以又稱乎此圓為歐拉圓。而第一個證明九點圓的人是彭賽列（1821年） 。1822年，卡爾·威廉·費爾巴哈也發現了九點圓，並得出「九點圓和三角形的內切圓和旁切圓相切」，因此德國人稱此圓為費爾巴哈圓，並稱這四個切點為費爾巴哈點。庫利奇與大上分別於1910年與1916年發表庫利奇－大上定理「圓周上四點任取三點做三角形 ，四個三角形的九點圓圓心共圓。」這個圓還被稱為四邊形的九點圓，此結果還可推廣到n邊形。

……

可以當成發想 的起點。假使設想姑且不論到底是怎樣『發現』的，且談已經『被發現』後，是否人們就能容易了解那些『關係』、『性質』、以及『證明』呢？有人說︰閱讀『證 明』容易，動手『證明』困難。似乎是講，既然都已理解了『證明』，焉有不曉『關係』與『性質』的耶！！若說條條大路通『羅馬』，就算盡觀了那些條條大路的 景緻，和『羅馬』之風光能夠彼此比較的嗎？？ ！！更何況『始、中、終』的『學習』循環不斷，舊『終』則啟新『始』布線織網深化『閱歷』。所以『學問』浸潤良久總有所悟，宛如說今日這門古老的『幾何學』，是以前那門新創之『幾何學』的嗎！！？？

何不讓我們舉個例子從『解析幾何』的觀點，來看一個歐式幾何之『證明』呢︰

解析幾何

Punktkoordinaten.svg

解析幾何（英語：Analytic geometry），又稱為坐標幾何（英語 ：Coordinate geometry）或卡氏幾何（英語：Cartesian geometry），早先被叫作笛卡兒幾何，是一種藉助於解析式進行圖形研究的幾何學分支。解析幾何通常使用二維的平面直角坐標系研究直線、圓、圓錐曲線、擺線、星形線等各種一般平面曲線，使用三維的空間直角坐標系來研究平面、球等各種一般空間曲面，同時研究它們的方程，並定義一些圖形的概念和參數。

在中學課本中，解析幾何被簡單地解釋為：採用數值的方法來定義幾何形狀，並從中提取數值的信息。然而，這種數值的輸出可能是一個方程或者是一種幾何形狀。

1637年，笛卡兒在《方法論》的附錄「幾何」中提出了解析幾何的基本方法。 以哲學觀點寫成的這部法語著作為後來牛頓和萊布尼茨各自提出微積分學提供了基礎。

對代數幾何學者來說，解析幾何也指（實或者複）流形，或者更廣義地通過一些複變數（或實變數）的解析函數為零而定義的解析空間理論。這一理論非常接近代數幾何，特別是通過讓-皮埃爾·塞爾在《代數幾何和解析幾何》領域的工作。這是一個比代數幾何更大的領域，不過也可以使用類似的方法。

【三角形三中線交於一點的證明】

三角形的中心

形心是三角形的幾何中心，通常也稱為重心，三角形的三條中線（頂點和對邊的中點的連線）交點，此點即為重心[1]。

220px-Triangle.Centroid.svg

三條中線共點證明

用西瓦定理逆定理可以直接證出：

\frac{BE}{EC} \cdot \frac{CF}{FA} \cdot \frac{AD}{DB}=\frac{1}{1} \cdot \frac{1}{1} \cdot \frac{1}{1}=1

因此三線共點。[2]

【前述證明所用之定理的證明】

塞瓦定理

塞瓦線段（cevian）是各頂點與其對邊或對邊延長線上的一點連接而成的直線段。塞瓦定理指出：如果\triangle ABC的塞瓦線段AD、BE、CF通過同一點O，則

\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB}=1

它的逆定理同樣成立：若D、E、F分別在\triangle ABC的邊BC、CA、AB或其延長線上（都在邊上或有兩點在延長線上），且滿足

\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB}=1，

則直線AD、BE、CF共點或彼此平行（於無限遠處共點）。當AD、BE、CF中的任意兩直線交於一點時，則三直線共點；當AD、BE、CF中的任意兩直線平行時，則三直線平行。

塞瓦定理2

它最先由義大利數學家喬瓦尼·塞瓦證明。

塞瓦定理1

證明

\because\quad\frac{BD}{DC}=\frac{\mathrm{S}_{\triangle ABD}}{\mathrm{S}_{\triangle ADC}}=\frac{\mathrm{S}_{\triangle OBD}}{\mathrm{S}_{\triangle ODC}}.

由等比性質,

\frac{BD}{DC}=\frac{\mathrm{S}_{\triangle ABD} - \mathrm{S}_{\triangle OBD}}{\mathrm{S}_{\triangle ADC} - \mathrm{S}_{\triangle ODC}}=\frac{\mathrm{S}_{\triangle ABO}}{\mathrm{S}_{\triangle CAO}}.
同理 \frac{CE}{EA}=\frac{\mathrm{S}_{\triangle BCO}}{\mathrm{S}_{\triangle ABO}},\;\frac{AF}{FB}=\frac{\mathrm{S}_{\triangle CAO}}{\mathrm{S}_{\triangle BCO}}.
\therefore\quad\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB}=\frac{\mathrm{S}_{\triangle ABO}}{\mathrm{S}_{\triangle CAO}} \cdot \frac{\mathrm{S}_{\triangle BCO}}{\mathrm{S}_{\triangle ABO}} \cdot \frac{\mathrm{S}_{\triangle CAO}}{\mathrm{S}_{\triangle BCO}}=1.

證畢。

【解析幾何的求解證明】

三角形重心

相異兩點 (x_0 , y_0) , \ (x_1, y_1) 決定一條線，這線的方程式可以寫成

y \ - y_0 \ = \frac{y_1 \ - y_0}{x_1 \ - x_0} \ (x \ - x_0)

相異三點可以決定一個三角形 \Delta \ ABC ，不失一般性，假設這三個頂點座標是 (0, 0) , \ (a, 0) , \ (b, c) ，那麼

線 L1 的方程式為

y \ = \ \frac{c}{2b \ - a} \ (2x \ - a)

線 L2 的方程式為

y \ = \ \frac{c}{b \ - 2a} \ (x \ - a)

線 L1 與線 L2 的交點 G ，求解聯立方程式可得

x \ = \ \frac{a + b} {3} \ , \ y \ = \ \frac{c}{3}

此『重心』之座標值，果然符合

幾何中心

中心分每條中線比為2:1，這就是說距一邊的距離是該邊相對頂點距該邊的1/3。如右圖所示：

如果三角形是由均勻材料做成的薄片，那麼幾何中心也就是質量中心。它的笛卡爾坐標是三個頂點的坐標算術平均值。也就是說，如果三頂點位於(x_a, y_a)，(x_b, y_b)，和(x_c, y_c)，那麼幾何中心位於：

\Big( \begin{matrix}\frac13\end{matrix} (x_a+x_b+x_c),\; \begin{matrix}\frac13\end{matrix} (y_a+y_b+y_c)\Big) = \begin{matrix}\frac13\end{matrix} (x_a, y_a) + \begin{matrix}\frac13\end{matrix} (x_b, y_b) + \begin{matrix}\frac13\end{matrix} (x_c, y_c)

詞條之所言。再者從向量

\vec{FG} \ = \ \frac{1}{6} (2b \ - a , 2c)

\vec{FC} \ = \ \frac{1}{2} (2b \ - a , 2c) \ = 3 \ \vec{FG}

之『關係』，故可知『重心』將『中線』分成了 2:1 之『性質』。

那麼所謂『證明』三角形三中線交於一點的事，也轉譯成了通過 A 點與 G 點的 L3 線，定然通過中點 D 。因此只需寫出 L3 的方程式，再驗證 D 點滿足那個參數方程式的耶！！如是將能夠體會不同論述之『難易煩簡』常十分不同的乎？？

─── 摘自《勇闖新世界︰ W!o《卡夫卡村》變形祭︰品味科學‧教具教材‧【專題】 PD‧箱子世界‧九點圓》

 

那人將能借閱讀『理解』設想『證明』 ── 幾何法、解析法 ── 『交比』的吧︰

Cross-ratio

In geometry, the cross-ratio, also called double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points A, B, C and D on a line, their cross ratio is defined as

(A,B;C,D)={\frac {AC\cdot BD}{BC\cdot AD}}

where an orientation of the line determines the sign of each distance and the distance is measured as projected into Euclidean space. (If one of the four points is the line’s point at infinity, then the two distances involving that point are dropped from the formula.)

The point D is the projective harmonic conjugate of C with respect to A and B precisely if the cross-ratio of the quadruple is −1, called the harmonic ratio. The cross-ratio can therefore be regarded as measuring the quadruple’s deviation from this ratio; hence the name anharmonic ratio.

The cross-ratio is preserved by the fractional linear transformations and it is essentially the only projective invariant of a quadruple of collinear points, which underlies its importance for projective geometry. In the Cayley–Klein model of hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio.

Cross-ratio had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who noted its key invariance property. It was extensively studied in the 19th century.[1] Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on the Riemann sphere.

Points A, B, C, D and A′, B′, C′, D′ are related by a projective transformation so their cross ratios, (A, B; C, D) and (A′, B′; C′, D′) are equal.

Terminology and history

Pappus of Alexandria made implicit use of concepts equivalent to the cross-ratio in his Collection: Book VII. Early users of Pappus included Isaac Newton, Michel Chasles, and Robert Simson. In 1986 Alexander Jones made a translation of the original by Pappus, then wrote a commentary on how the lemmas of Pappus relate to modern terminology.[2]

Modern use of the cross ratio in projective geometry began with Lazare Carnot in 1803 with his book Géométrie de Position. The term used was le rapport anharmonique (Fr: anharmonic ratio). German geometers call it das Doppelverhältnis (Ger: double ratio).

Given three points on a line, a fourth point that makes the cross ratio equal to minus one is called the projective harmonic conjugate. In 1847 Carl von Staudt called the construction of the fourth point a Throw (Wurf), and used the construction to exhibit arithmetic implicit in geometry. His Algebra of Throws provides an approach to numerical propositions, usually taken as axioms, but proven in projective geometry.[3]

The English term “cross-ratio” was introduced in 1878 by William Kingdon Clifford.[4]

D is the harmonic conjugate of C with respect to A and B, so that the cross-ratio (A, B; C, D) equals −1.

Definition

The cross-ratio of a 4-tuple of distinct points on the real line with coordinates z1, z2, z3, z4 is given by

{\displaystyle (z_{1},z_{2};z_{3},z_{4})={\frac {(z_{3}-z_{1})(z_{4}-z_{2})}{(z_{3}-z_{2})(z_{4}-z_{1})}}.}

It can also be written as a “double ratio” of two division ratios of triples of points:

{\displaystyle (z_{1},z_{2};z_{3},z_{4})={\frac {z_{3}-z_{1}}{z_{3}-z_{2}}}:{\frac {z_{4}-z_{1}}{z_{4}-z_{2}}}.}

The cross-ratio is normally extended to the case when one of z1, z2, z3, z4 is infinity  \infty , this is done by removing the corresponding two differences from the formula.

For example: if  {\displaystyle z_{1}=\infty } the cross ratio becomes:

{\displaystyle (z_{1},z_{2};z_{3},z_{4})={\frac {(z_{3}-\infty )(z_{4}-z_{2})}{(z_{3}-z_{2})(z_{4}-\infty )}}={\frac {(z_{4}-z_{2})}{(z_{3}-z_{2})}}.}

In geometry, if A, B, C and D are collinear points, then the cross ratio is defined similarly as

{\displaystyle (A,B;C,D)={\frac {AC\cdot BD}{BC\cdot AD}},}

where each of the distances is signed according to a consistent orientation of the line.

The same formulas can be applied to four different complex numbers or, more generally, to elements of any field and can also be extended to the case when one of them is the symbol ∞, by removing the corresponding two differences from the formula. The formula shows that cross-ratio is a function of four points, generally four numbers z_{1},\ z_{2},\ z_{3},\ z_{4} taken from a field.

Projective geometry

Cross-ratio is a projective invariant in the sense that it is preserved by the projective transformations of a projective line.

In particular, if four points lie on a straight line L in R2 then their cross-ratio is a well-defined quantity, because any choice of the origin and even of the scale on the line will yield the same value of the cross-ratio.

Furthermore, let {Li | 1 ≤ i ≤ 4} be four distinct lines in the plane passing through the same point Q. Then any line L not passing through Q intersects these lines in four distinct points Pi (if L is parallel to Li then the corresponding intersection point is “at infinity”). It turns out that the cross-ratio of these points (taken in a fixed order) does not depend on the choice of a line L, and hence it is an invariant of the 4-tuple of lines {Li}.

This can be understood as follows: if L and L′ are two lines not passing through Q then the perspective transformation from L to L′ with the center Q is a projective transformation that takes the quadruple {Pi} of points on L into the quadruple {Pi′} of points on L′.

Therefore, the invariance of the cross-ratio under projective automorphisms of the line implies (in fact, is equivalent to) the independence of the cross-ratio of the four collinear points {Pi} on the lines {Li} from the choice of the line that contains them.

 

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