The name "Butterfly Theorem" first appeared in the February issue of the American Mathematical Monthly (1944), because the figure of the topic is like a butterfly. The name is not only similar, but also similar. Since its birth, there are at least dozens of ways to prove this problem, and the discussion on this issue has never stopped, and many beautiful and enlightening by-products have been obtained. This dancing butterfly, from elementary to advanced, has flown over many fields of mathematics, and its wings are covered with colorful fragrance and flowers.
The butterfly theorem is expressed as follows: as shown in the following figure, m is the midpoint of the chord AB on the circle O, and the connecting lines DE and CF intersect AB at P and Q respectively. Verify PM=QM.
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Let's discuss one of Mr Denton's proofs. This proof has been regarded as a classic proof of plane geometry including analytic geometry since its birth. Only a few lines of proof are concise, but they are informative. At first glance, it still feels a bit abstract, so I need to sort out my thoughts and add some comments. The study of Confucian classics and historiography in ancient China was almost hermeneutics from the beginning. Later, with the gradual deepening of secularization, there are more and more appreciation works as a medium between creators and audiences, such as Wen Han Zhang Ci. Such a classic proof is worth appreciating.
Firstly, a rectangular coordinate system with AB straight line as X axis, OM straight line as Y axis and M as origin is established. For convenience, we might as well set the circle as a unit circle. The coordinate o (0, -a) of the center.
Imagine that if we can know the equation of the straight line where ED and CF are located and get the abscissa of P and Q, the problem will be solved. E, d, c, f are the intersection of the graph composed of two straight lines EF and CD intersecting m-let's call it (EF * CD)-and a circle O. Write the equation of (ef * CD) first.
Let the equation of EF's straight line be y=k 1x, and that of CD's straight line be y=k2x. If we are sensitive to addition in Boolean algebra, a concrete example is logic circuit, and we can soon get that the equation corresponding to graph (EF*CD) is (y-K 1x) * (y-K2x) = 0. Because for any point on the diagram, substituting this expression can be established.
Let's consider the intersection of a graph (EF*CD) and a circle. If we look back at the equation of the conic system passing through the intersection, it is not difficult to get that the equation of the conic system passing through the intersection of (EF*CD) and the circle is [x 2+(y+a) 2-1]+? λ (y-k1x) * (y-k2x) = 0. Intuitively, there must be a point on the graph that can make both parts zero at the same time, so let the whole be zero, that is, the graph passes through the intersection of (EF*CD) and the circle. Multiplication in logic circuits.
The above are some necessary precautions. If you understand this, the proof process will be very simple and beautiful:
from[x ^ 2+(y+a)2- 1]+？ λ(y—k 1x)*(y—k2x)=0，
? Let y=0, and get (1+λ k1k2) x2+a2-1= 0.
It can be seen that the coefficient of the first term of x is 0. Proved by Vieta theorem xP+xQ=0, namely PM=QM.
It should be noted that there are six groups of straight lines passing through E, D, C and F, and the conclusion to be proved is one of them. CD and EF intersect at the origin and naturally coincide. As shown in the figure, the lines cut by the extension lines of CE and DF and AB are also equal.
Mr. Denton's classic proof is concise and beautiful, rich in meaning and full of holistic view. The application of skills can be called four or two thousand pounds, and mathematical thoughts are integrated with it, such as antelope hanging horns, which is like an inexhaustible short poem, such as Goethe's "Nocturne of the Wanderer".
Let's make some extensions. Butterfly theorem is not only applicable to circles, but also to other quadratic curves such as ellipses, hyperbolas, parabolas and even degenerate two intersecting lines. It is also convenient to popularize with the above proof method.
We only need to set the general formula X 2+Bxy+Cy 2+DX+EY+F = 0 of the conic. Establish the above coordinate system, and substitute a (-m, 0) and b A(-m, 0) to get f =-m 2 and d=0. That is, x 2+bxy+cy 2+ey-m 2 = 0. Construct x 2+bxy+cy 2+ey-m 2+? λ (y-K 1x) * (y-K2x) = 0, which proves exactly the same.
Can we look at the above promotion from a higher mathematical perspective? This requires entering the field of projective geometry. First prove the butterfly theorem in a circle with projective geometry.
As shown in the above figure, the relationship between the intersection ratio of four collinear straight lines and the intersection ratio of four collinear ordered points is: (abcd)=(APMB), (a'b'c'd')=(AMQB). Because every circumferential angle subtended by an arc in a circle is equal, two groups of four straight lines are "coincident" and obviously have the same cross ratio. Then (APMB)=(AMQB).
According to the definition of cross ratio, (MA/MP)? (BP/BA)=(QA/QM)？ (BM/BA). From MA=MB, BP/MP=QA/QM is obtained after reduction. Subtract 1 from both sides, and BM/MP=MA/QM. From MA=MB, it can be proved that PM=QM.
The point is not the proof process itself. What I want to emphasize in this paper is that because a conic is only the projection of a circle, any property of the circle that is unchanged under the projection will also be owned by any conic. In this example, the cross ratio is invariant under projective transformation, so if the circle is projected into an arbitrary quadratic curve, the equation (abcd)=(a'b'c'd') still holds. The following proof process is of course exactly the same. For any conic, this seems to be an amazing conclusion, but it is so easy to get.
Irelan's root program, which is familiar to math lovers, tells us that a geometry studies nothing more than invariance under a certain transformation, and the importance of topology lies in that it studies invariance under the most drastic transformation. In the example of butterfly theorem, we have deepened our understanding of this important mathematical idea through the concrete example of the invariance of cross ratio under projective transformation. On the contrary, with such a macro perspective, it is so easy to solve the problem of extending butterfly theorem to ordinary cones.
Yellow butterflies in August, hovering in the grass of our West Garden in pairs. I especially like these two poems in Li Bai's Long March. On a cloudy afternoon in late spring, I mentioned the famous butterfly theorem again, and used a beautiful cone system and projective geometry to fly together, flying over many beautiful scenery.