Please imagine independent and different sizes of three circles.

If we draw two tangent lines concerning each pair of two circles, the three points of intersection of each two tangent lines will stand in the same straight line.

However, why does it occur?

If we draw two tangent lines concerning each pair of two circles, the three points of intersection of each two tangent lines will stand in the same straight line.

However, why does it occur?

Here, I'm going to show the way to solve this mathematical problem by vector.

At first, I define , and as a vector which starts at the center of each circle and finishes at the center of another circle like diagram below.

Next, I define and as a vector which starts at the point of intersection of two tangent of particular set of two circles and finishes different point of the similar intersection.

And, I define radius of each circle as , and .

The important point of this problem is to describe by , that is, to prove that is on the same line as .

Then, how we can describe by .

Please think about describe and in different formula, that is, the formula using , , , , and because if we can describe and by same components, we may find new relationship between and .

At first, I define , and as a vector which starts at the center of each circle and finishes at the center of another circle like diagram below.

Next, I define and as a vector which starts at the point of intersection of two tangent of particular set of two circles and finishes different point of the similar intersection.

And, I define radius of each circle as , and .

The important point of this problem is to describe by , that is, to prove that is on the same line as .

Then, how we can describe by .

Please think about describe and in different formula, that is, the formula using , , , , and because if we can describe and by same components, we may find new relationship between and .

When we attentively stare at two circles, there we can find similar triangles around two circles.

The ratio of same side of similar triangles is same in any side.

Therefore, the ratio of the length of the vectors on the same line ( and ) is same as the ratio of the radius of each circle ( and ).

The ratio of same side of similar triangles is same in any side.

Therefore, the ratio of the length of the vectors on the same line ( and ) is same as the ratio of the radius of each circle ( and ).

When we apply this to the and , they are described finally like this (details of calculation is below).

This vector equilibrium indicates that is on the same line of .

Then, problem is solved.

In addition, I think it is also interesting that the ratio of and is only dependent on the radius of each circle.

Then, problem is solved.

In addition, I think it is also interesting that the ratio of and is only dependent on the radius of each circle.

I knew this mathematical problem in the magazine "Science" in the middle of 1970s.

At that time, I had learned only algebra.

Therefore, the simple and elegant solution which I explained before the vector solution was very shocking to me.

This mathematical problem which I accidentally encountered left me important teachings that complicated problem can be solved easily by thinking from another dimension.

I always hold this teaching in my mind.

SDGs ends its declaration by the words that "Leave No One Behind".

To realize the world which SDGs aims at, all countries must endeavor and trust each other profoundly.

In the future world beyond 2030, which is the deadline of SDGs, people will have changed their relationship to the nature.

I hope that people will no longer try to "develop" nature, but try to walk with nature.

If we try hard from now, the world beyond SDGs will close to the stable prosperity in the peaceful harmony with nature.

In addition, I also hope that people will stop aiming at "expanding equilibrium" and start aiming at "contract equilibrium" which will keep global population optimum and bring global peace.

Japanese sake industry must reconsider how they should exist from the new dimension that how we will be able to build and hand over sustainable world to our children.

In the present sake industry, there are undesirable actions such as exploitation of distilled alcohol from foreign countries or easygoing production of sake with no consideration about food and energy waste.

To build sustainable world for future people including our children, sake industry also should introduce the idea of Circular Economy which reduce the waste of resources as long as possible.

I hope that these new action in sake industry will stimulate other industry in the end.

Satoshi Miyajima,

Owner of the locally rooted sake brewery SHINANO-NISHIKI

( Miyajima Brewery Co. )

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( Miyajima Brewery Co. )

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Hobbies are science, climbing, star watching & photography.

Motto: Too much is as bad as too less.

Motto: Too much is as bad as too less.

SDGs "To the Future Beyond"

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