How to prove the butterfly theorem, the butterfly theorem is the easiest to prove

The simplest proof of the butterfly theorem is as follows:

1. M as the intersection point of the inner chord of the circle is unnecessary and can be moved outside the circle.

2. The circle can be changed to any conic curve.

3. Turn the circle into a zither shape, with m as the diagonal intersection.

4. Removing the condition of the midpoint, the conclusion becomes a proportional formula for the general directed line segment, called "Kandi's theorem", which is not satisfied when the midpoint is not satisfied. This pair holds for both 1,2.

The butterfly theorem is one of the most brilliant results of ancient Euclidean plane geometry. This proposition first appeared in 1815 and was proved by Horner. The name "butterfly theorem" first appeared in the February 1944 issue of the American Mathematical Monthly, and the figure of the title resembles a butterfly.

The list of proofs of this theorem is endless, and it is still studied by math enthusiasts, and there are various variations from time to time during exams.

This proposition first appeared as a solution problem in 1815 A.D. in the British magazine "gentleman's diary'S Diary) pp. 39-40 (p39-40). Interestingly, until 1972, people's proofs were not rudimentary and cumbersome.

In the year this article was published, Horner, a self-taught secondary school mathematics teacher in England (who invented the Horner method of approximating the roots of polynomial equations), gave the first proof, which was completely rudimentary; Another proof was given by Richard Taylor.

Another kind of early proof by MIt is given in a book by Mile Brand in 1827. The most concise method is the one of projective geometry, which was developed by the British J Kaishi"a sequel to the first six books of the elements of euclid"Given, there is only one sentence, which is the ratio of the wiring harness.

The name "butterfly theorem" first appeared in the February 1944 issue of the American Mathematical Monthly, and the title depicts a butterfly.

The basic formula of the butterfly model is: ad:bc=oa:

OC, the perturbation of the butterfly theorem is one of the most brilliant results of ancient Euclidean plane geometry. This proposition first appeared in 1815 and was proved by W. G. Horner.

The name "butterfly theorem" first appeared in the American Mathematical Monthly

In the February 1944 issue, the title is shaped like a butterfly. There are many ways to prove this theorem, and it is still studied by math enthusiasts, and there are various variations in exams.

Butterfly theorem: Let m be the midpoint of the inner chord PQ of the circle, and pass M to be the chord ab and cd. Let AD and BC each intersect Pq at the points X and Y, then M is the midpoint of Xy.

Removing the midpoint of the strip modification, the conclusion becomes a general proportional formula about directed line segments, called "Kandi's theorem", which satisfies when not the midpoint: 1 my-1 mx=1 mq-1 mp, which holds both 2 and 3.

Brief introduction. The butterfly theorem was first published in 1815 in a popular magazine, The Men's Diary, as a proven-proof problem, and was named because of its peculiar geometric figure that resembles a butterfly.

There have been many beautiful and strange solutions in history, the earliest of which should be the non-elementary proof given by Horner. As for the proof method of elementary mathematics, in foreign sources, it is generally believed that it was first proposed by Stewin, a mathematics teacher at Xingkut Middle School in Yudong, and he gave the proof of the area method.