Six points of Japanese yen

Six points of Japanese yen

Six points of Japanese yen (ろくてんえん) is Japanese yen to go along the foot of six other perpendicular lines which I took down to two sides in total from the foot of the perpendicular line which I brought down from each triangular top. A theorem that these six points are on the same circumference is called "a theorem of six points of Japanese yen".

Because Henry Martin Taylor (Henry Martin Taylor,1842-1927) announced the article about this Japanese yen in the 1880s, a name called Madoka Taylor is common in Europe and America.

Table of contents

Proof

The proof of the existence of this Japanese yen is possible by geometric knowledge to learn by a junior high school.

In A,B,A',B', the same circumference is located above from ,∠ AA'B= ∠ AB'B in an upper figure. Thus, it is ⊿ CBA ∽⊿ CB'A' in being ∠ CBA= ∠ CB'A'. It is ⊿ CA'B' ∽⊿ CA₂B₁ equally.

⊿From CA'H ∽⊿ CC₂C', ⊿ CB'H ∽⊿ CC₁C' CA': CC₂=CH:CC'=CB': CC₁ stops by; and ⊿ CA'B' ∽⊿ CC₂C₁.

⊿Because it is CA₂B₁ ∽⊿ CC₂C₁, it is ∠ CA₂B₁= ∠ CC₂C₁, and, as for four points of A₂,B₁,C₁,C₂, the same circumference is located above. Similarly, four points of A₁,B₁,B₂,C₂ are on the same circumference, too.

⊿C₁B₂ is parallel to CB than CBA ∽⊿ C'B'A ∽⊿ C₁B₂A.

⊿Four points of B₁,B₂,C₁,C₂ are on the same circumference than ∠ CC₂C₁= ∠ B₂B₁B= ∠ B₁B₂C₁ this from CBA ∽⊿ CB'A' ∽⊿ CC₁C₂, ⊿ CBA ∽⊿ C'BA' ∽⊿ B₂BB₁, too.

Six points were shown to be on the same circumference by the above.

The center and radius

In the center of six points of Japanese yen, the straight line same as reservedness, ルモワーヌ point and others is located above.

When the radius assumes R, the size of three interior angles α, β, γ at a circumradius,

${\displaystyle R_{T}=R{\sqrt {\sin ^{2}\alpha \sin ^{2}\beta \sin ^{2}\gamma +\cos ^{2}\alpha \cos ^{2}\beta \cos ^{2}\gamma}}}$ 

I can express で.

Allied item

• Nine-point circle

Outside link

• Darij Grinberg and Eric W. Weisstein, "Taylor Circle" - MathWorld. (English)

It is acquired by "https://ja.wikipedia.org/w/index.php?title= six points Japanese yen &oldid=55870980"

This article is taken from the Japanese Wikipedia Six points of Japanese yen

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