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チェバの定理

この切手シートは、2016年第57回国際数学オリンピック香港大会を記念して香港郵政が発行したものです(下記参照)。

問題

$\triangle ABC$ の各辺 $AB$、$BC$、$CA$ 上にそれぞれ点 $Z$、$X$、$Y$ をとる。 3つの線分 $AX$、$BY$、$CZ$ が1点 $O$ で交わるとき、

$$\frac{AZ}{ZB}\cdot\frac{BX}{XC}\cdot\frac{CY}{YA} = 1$$

である。

この定理を使っても重心が存在することを示せますね。 2点、例えば $X$ と $Y$ が中点なら $Z$ も中点になりますから、交点 $O$ は重心です。

57th International Mathematical Olympiad 2016 Special Stamps

Date of Issue: 6 July 2016

The 57th International Mathematical Olympiad (IMO), to be held in Hong Kong from 6 to 16 July 2016, is hosted by the International Mathematical Olympiad Hong Kong Committee (IMOHKC) with the Hong Kong University of Science and Technology as the Host University and the Education Bureau as the Supporting Organisation. More than 100 teams from all over the world will compete in this year’s contest. To mark this major mathematical event, Hongkong Post issues a stamp sheetlet on this theme.

The IMO is a competition for secondary school students. The event, held annually in a different country or territory, provides an opportunity for exchange among youngsters from around the world who are under the age of 20 and gifted in mathematics, and nurtures students’ interest in mathematics. The IMOHKC was founded in 1986. In collaboration with the Education Bureau and the Hong Kong Academy for Gifted Education, the committee voluntarily provides talented local students with mathematical Olympiad training and other learning opportunities. Hong Kong has been participating in the IMO since 1988, with the six representing contestants handpicked annually by the IMOHKC among the students attending training courses.

The stamp sheetlet features an illustration of Ceva’s Theorem, a tool frequently used in mathematics to solve problems in geometry. The stamp is circular in shape and shows a geometric problem formulated by Hong Kong, which was adopted as one of the questions in the 2010 IMO.

Acknowledgement: International Mathematical Olympiad Hong Kong Committee

2010年第51回国際数学オリンピックカザフスタン大会(問題2)

$$\angle BAF = \angle CAE < \frac{1}{2} \angle BAC$$

References

Last-modified: 2016-09-29 (木) 15:12:16 (2098d)