There have been many beautiful and strange solutions, the earliest of which should be Horner's proof in 8 15. As for the proof method of elementary mathematics, in foreign materials, it is generally believed that Steven, a middle school teacher, first put forward the proof method of area, in which the area formula is applied: S= 1/2.

BCSINA. 1985, in the first issue of Henan Mathematics Teacher, Comrade Du Xilu introduced the butterfly theorem to China with the title "Famous Problems in Plane Geometry and Their Wonderful Solutions", and since then, the butterfly theorem has spread all over China.

This paper introduces a simple method to prove elementary mathematics.

It is proved that the center O is the vertical line between AD and B, and the vertical foot is S and T, connecting OX, OY and OM. SM .MT .

∫△SMD∽△CMB and SD= 1/2ADBT= 1/2BC,

∴DS/BT=DM/BM and ≈ D = ∠ B.

∴△MSD∽△MTB,∠MSD=∠MTB

∴∠msx=∠mty； And o, s, x, m and o, t. Y .m are a four-point * * * circle,

∴∠XOM=∠YOM

∵OM⊥PQ∴XM=YM

As shown in figure 1, the major axis A 1A2 of the ellipse is parallel to the X axis, and the minor axis B 1B2 is on the Y axis with the center of M(o, r) (b > r > 0).

(1) Write the equation of ellipse, and find the focal coordinates and eccentricity of ellipse;

(2) the straight line y=kx intersects the ellipse at two points c (x 1, y 1) and d (x2, y2) (y2 > 0); The straight line y=k2x intersects the ellipse at two points G(x3, y3) and H(x4, y4) (y4 > 0).

Verification: k1x1x 2/(x1+x2) = k2x3x4/(x3+x4)

(Ⅲ) For C, D, G and H in (Ⅱ), let CH intersect with X axis at point P and GD intersect with X axis at point Q. ..

Verification:

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Too strong; administrator

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=

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OQ

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(The proof process does not consider the case that CH or GD is perpendicular to the X axis. )

I can't solve the butterfly theorem with quadratic function.

Maybe you're on your own. Come on! ! ! !