Proof that the shortest edge length of a Kepler right triangle is equal in measure to the circumference of a circle with a diameter that is equal in measure to 1 quarter of the second longest edge length of the Kepler right triangle:
The shortest edge length of the Kepler right triangle is 12.
If the hypotenuse of a Kepler right triangle is divided by the measure for the shortest edge length of the Kepler right triangle the result is the Golden ratio of cosine (36 degrees) multiplied by 2 = Phi = 1.618033988749895.
The ratio of the hypotenuse of a Kepler right triangle divided by the second longest edge length of the Kepler right triangle is also the square root of the Golden ratio = √φ = 1.272019649514069.
The shortest edge length of the Kepler right triangle is 12.
12 multiplied by the Golden ratio of cosine (36 degrees) multiplied by 2 = Phi = 1.618033988749895 = (12 times (φ)) = 19.41640786499874.
The hypotenuse of a Kepler right triangle that has its shortest edge length equal to 12 is equal to 12 times (φ) = the Golden ratio = 19.41640786499874.
Apply the Pythagorean theorem to the hypotenuse of the Kepler right triangle and the shortest edge length of the Kepler right triangle to get the measure for the second longest edge length of the Kepler right triangle.
12 times φ = 19.41640786499874 squared = 376.996894379984929.
12 squared = 144.
(12 times φ times 12 times φ subtract 144) = 12 times φ times 12 times φ = 376.996894379984929 subtract 144 = 232.996894379984929.
The square root of 232.996894379984929 = 12 times √φ = 15.26423579416883.
The second longest edge length of the Kepler right triangle that has its shortest edge length equal to 12 is equal to 12 times √φ = 15.26423579416883.
12 times √φ = 15.264235794168832 divided by 4 = 3 times √φ = the square root of the Golden ratio = 3.816058948542208 = the diameter of the circle.
3 times √φ = 15.264235794168832 divided by the square root of the Golden ratio = 1.272019649514069 = 12.
The shortest edge length of the Kepler right triangle has been divided into 12 equal units of measure and is the same measure as the circumference of a circle with a diameter that is equal in measure to 1 quarter of the second longest edge length of the Kepler right triangle.
12 is the numerical value for both the circumference of the circle and the shortest edge length of the Kepler right triangle that has its hypotenuse equal to 12 times (φ) = 19.41640786499874 while the second longest edge length of the Kepler right triangle is equal to 12 times √φ = 15.264235794168832.
Circumference of circle and the shortest edge length of the mentioned Kepler right triangle are both equal to 12.
1 quarter of the second longest edge length of the Kepler right triangle and the diameter of the circle are both equal to 3 times the square root of the Golden ratio = (3 times √φ) = 3.816058948542208.
The Golden ratio = the square root of 5 plus 1 divided by 2 = 1.618033988749895 or alternatively cosine (36 degrees) multiplied by 2 = (φ) = (√(5) plus 1)/2 = 1.618033988749895.
The square root of the Golden ratio = √φ = 1.272019649514069.
The square root of the Golden ratio = √φ = 1.272019649514069 multiplied by 3 = the ratio (3 times √φ) = 3.816058948542207.
The ratio (3 times √φ) = 3.816058948542207 is the square root of the Golden ratio √φ = 1.272019649514069 multiplied by 3 and also 1 quarter of the second longest edge length of a Kepler right triangle that has its shortest edge length divided into 12 equal units of measure.
12 divided by (3 times √φ) = 3.816058948542208 = 𝝿 = Pi = 4/√φ = 3.144605511029692.
Remember to multiply the shortest edge length of the Kepler right triangle that is 12 by cosine (36 degrees) multiplied by 2 = (φ) = The Golden ratio = Phi = (φ) = (√(5) plus 1)/2 = 1.618033988749895 to get the measure for the hypotenuse of the Kepler right triangle = the ratio 12 times (φ) = 19.41640786499874 then apply the Pythagorean theorem to the shortest edge length of the Kepler right triangle and the hypotenuse of the Kepler right triangle to get the measure for the second longest edge length of the Kepler right triangle.
The diameter of the circle is equal to 1 quarter of the second longest edge length of the Kepler right triangle.
The shortest edge length of the Kepler right triangle is equal in measure to the circumference of the circle that has a diameter that is equal in measure to 1 quarter of the second longest edge length of the Kepler right triangle.
Divide the measure of the shortest edge length of the Kepler right triangle by the measure for 1 quarter of the second longest edge length of the Kepler right triangle to get 𝝿 = Pi = 4/√φ = 3.144605511029693144.
Pi can also be calculated from the diagram of a circle contained inside of a square if the width of the square is the same measure as the diameter of the circle because if the perimeter of a square is divided by the square root of the Golden ratio = √φ = 1.272019649514069 then the result is the measure for the circumference of the circle that has a diameter that is the same measure as the width of a square.
Example:
The width of the square = 3 times the square root of the Golden ratio = (3 times √φ) = 3.816058948542208.
Diameter of the circle that is contained inside of the square = 3 times the square root of the Golden ratio = (3 times √φ) = 3.816058948542208.
3 times the square root of the Golden ratio = (3 times √φ) = 3.816058948542208 multiplied by 4 = (12 times √φ) = the ratio 15.264235794168832.
Perimeter of square that contains the circle that has a diameter that is the same measure as the width of the square = (3 times √φ) = the ratio 15.264235794168832.
12 times the square root of the Golden ratio = the ratio = (12 times √φ) = 15.264235794168832 divided by the square root of the Golden ratio = √φ = 1.272019649514069 = 12.
12 is the measure for the circumference of the circle that has a diameter that is the same measure as the width of the square.
12 divided by 3 times the square root of the Golden ratio = (3 times √φ) = 3.816058948542208 = 𝝿 = Pi = 4/√φ = 3.144605511029693144.
(3 times √φ) = 3.816058948542207.
(12 times √φ/(4)) = 3.816058948542207.
(12/(3 times √φ)) = 4/√φ = 3.144605511029693144.
(12/(12 times √φ/(4))) = 4/√φ = 3.144605511029693144.
The true value of 𝝿 = Pi = 4/√φ = 3.144605511029693144.
Proof that the second longest edge length of a Kepler right triangle is equal in measure to the circumference of a circle with a diameter that is equal in measure to 1 quarter of the hypotenuse of the Kepler right triangle:
The shortest edge length of the Kepler right triangle is 12.
If the hypotenuse of a Kepler right triangle is divided by the measure for the shortest edge length of the Kepler right triangle the result is the Golden ratio of cosine (36 degrees) multiplied by 2 = Phi = 1.618033988749895.
The ratio of the hypotenuse of a Kepler right triangle divided by the second longest edge length of the Kepler right triangle is also the square root of the Golden ratio = √φ = 1.272019649514069.
The shortest edge length of the Kepler right triangle is 12.
12 multiplied by the Golden ratio of cosine (36 degrees) multiplied by 2 = Phi = 1.618033988749895 = (12 times (φ)) = 19.41640786499874.
The hypotenuse of a Kepler right triangle that has its shortest edge length equal to 12 is equal to 12 times (φ) = the Golden ratio = 19.41640786499874.
Apply the Pythagorean theorem to the hypotenuse of the Kepler right triangle and the shortest edge length of the Kepler right triangle to get the measure for the second longest edge length of the Kepler right triangle.
12 times (φ) = 19.41640786499874 squared = 376.996894379984929.
12 squared = 144.
(12 times φ times 12 times φ subtract 144) = 12 times φ times 12 times φ = 376.996894379984929 subtract 144 = 232.996894379984929.
The square root of 232.996894379984929 = 12 times √φ = 15.26423579416883.
The second longest edge length of the Kepler right triangle that has its shortest edge length equal to 12 is equal to 12 times √φ = 15.26423579416883.
The second longest edge length of the Kepler right triangle is equal to 12 equal units of measure.
The second longest edge length of the mentioned Kepler right triangle is the same measure as the circumference of a circle with a diameter that is equal in measure to 1 quarter of the hypotenuse the Kepler right triangle.
12 times √φ = 15.26423579416883 is the numerical value for both the circumference of the circle and the second longest edge length of the Kepler right triangle that has its hypotenuse equal to 12 times (φ) = 19.41640786499874 while the shortest edge length of the Kepler right triangle is equal to 12.
Circumference of circle is equal to 12 times is √φ = 15.26423579416883 and the shortest edge length of the mentioned Kepler right triangle is equal to 12.
1 quarter of the hypotenuse of the Kepler right triangle and the diameter of the circle are both equal to 3 times (φ) = 4.854101966249685.
12 times (φ) = 19.416407864998738.
3 times (φ) = 4.854101966249685.
(3 times (φ) times 4) = 12 times (φ) = 19.416407864998738.
The Golden ratio = the square root of 5 plus 1 divided by 2 = 1.618033988749895 or alternatively cosine (36 degrees) multiplied by 2 = (φ) = (√(5) plus 1)/2 = 1.618033988749895.
√φ = The square root of the Golden ratio = √φ = 1.272019649514069.
The square root of the Golden ratio = √φ = 1.272019649514069 multiplied by 3 = the ratio (3 times √φ) = 3.816058948542207.
The ratio (3 times √φ) = 3.816058948542207 is the square root of the Golden ratio √φ = 1.272019649514069 multiplied by 3 and also 1 quarter of the second longest edge length of a Kepler right triangle that has its shortest edge length divided into 12 equal units of measure.
12 divided by (3 times √φ) = 3.816058948542208 = 𝝿 = Pi = 4/√φ = 3.144605511029692.
Remember to multiply the shortest edge length of the Kepler right triangle that is 12 by cosine (36 degrees) multiplied by 2 = (φ) = The Golden ratio = Phi = (φ) = (√(5) plus 1)/2 = 1.618033988749895 to get the measure for the hypotenuse of the Kepler right triangle = the ratio 12 times (φ) = 19.41640786499874 then apply the Pythagorean theorem to the shortest edge length of the Kepler right triangle and also the hypotenuse of the Kepler right triangle to get the measure for the second longest edge length of the Kepler right triangle.
The diameter of the circle is equal to 1 quarter of the hypotenuse of the Kepler right triangle.
The second longest edge length of the Kepler right triangle is equal in measure to the circumference of the circle that has a diameter that is equal in measure to 1 quarter of the hypotenuse of the Kepler right triangle.
Divide the measure of the second longest edge length of the Kepler right triangle by the measure for 1 quarter of the hypotenuse of the Kepler right triangle to get 𝝿 = Pi = 4/√φ = 3.144605511029693144.
Pi can also be calculated from the diagram of a circle contained inside of a square if the width of the square is the same measure as the diameter of the circle because if the perimeter of a square is divided by the square root of the Golden ratio = √φ = 1.272019649514069 then the result is the measure for the circumference of the circle that has a diameter that is the same measure as the width of a square.
Example:
The width of the square = 3 times the square root of the Golden ratio = (3 times √φ) = 3.816058948542208.
Diameter of the circle that is contained inside of the square = 3 times the square root of the Golden ratio = (3 times √φ) = 3.816058948542208.
3 times the square root of the Golden ratio = (3 times √φ) = 3.816058948542208 multiplied by 4 = (12 times √φ) = the ratio 15.264235794168832.
Perimeter of square that contains the circle that has a diameter that is the same measure as the width of the square = (3 times √φ) = the ratio 15.264235794168832.
12 times the square root of the Golden ratio = the ratio = (12 times √φ) = 15.264235794168832 divided by the square root of the Golden ratio = √φ = 1.272019649514069 = 12.
12 is the measure for the circumference of the circle that has a diameter that is the same measure as the width of the square.
12 divided by 3 times the square root of the Golden ratio = (3 times √φ) = 3.816058948542208 = 𝝿 = Pi = 4/√φ = 3.144605511029693144.
(3 times √φ) = 3.816058948542207.
(12 times √φ/(4)) = 3.816058948542207.
(12 times √φ/(3 times (φ)) = 4/√φ = 3.144605511029693144.
(12/(12 times √φ/(4))) = 4/√φ = 3.144605511029693144.
The true value of 𝝿 = Pi = 4/√φ = 3.144605511029693144.
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Example of proof 1:
https://drive.google.com/file/d/1eS-HGAd_O3ZkmeMiWS-u81X2uRBOY82E/view?usp=sharing
Example of proof 2:
https://drive.google.com/file/d/10RjD1JEUzrbvJGiEvaIa2ONadxnDeeKc/view?usp=sharing
Example of proof 3:
https://drive.google.com/file/d/1bxiXiJpxFY2BSJtcv4b9-aSR3uduROBU/view?usp=sharing
Geometric diagram scan:
https://drive.google.com/file/d/1XeYiSO8ls2vAnGOkxCdf82KnxXUv507y/view?usp=sharing
