Decimal Fractional Parts of Segment Dim's on Hypotenuses of Dissimilar Right Triangles

DECIMAL FRACTIONS:
The decimal, fractional part of a positive real number is the excess beyond that number's integer part; e.g., for the number 3.75, the numbers to the right of the decimal point make up the decimal, fractional part of the positive real number 3. The decimal fractional part, .75 equals the fraction 3/4.

For sources, see links below;

https://en.wikipedia.org/wiki/Decimal?wprov=sfla1.

https://en.wikipedia.org/wiki/Fractional_part?wprov=sfla1.

EXPLANATION:
Dimensions with decimal, fractional parts invariably occur on the two segments of the hypotenuse of a right triangle, caused by the altitude to the hypotenuse. The product of these segments equals the square of the altitude to the hypotenuse.

For further information on identical altitudes to the hypotenuse in 3 different right triangles, see link to Quora Blog below, and Items 15E thru 15H.

https://righttrianglecuriosities.quora.com/1-Altitude-To-The-Hypotenuse….

In the link below, see altitude to hypotenuse h and segments "p" and "q".

https://en.wikipedia.org/wiki/Geometric_mean_theorem?wprov=sfla1.

NOTE: Segments "p" and "q" are used in this post to help visualize graphically their association with actual dimensions calculated here for the lesser and greater segments on the hypotenuse of three dissimilar right triangles.

For example, hypotenuse segments; SA1, SB1 & SC1 on dissimilar right triangles A, B & C, correspond to the greater segment "q", and segments SA2, SB2 & SC2 correspond to the lesser segment "p".

ABOUT THIS POST; This post deals with the incredibly obscure, and recurring identical decimal fractions, of actual calculated dimensions of the two closed linesegments on the hypotenuses of 3 dissimilar right triangles, A, B & C, caused by the same altitude to the hypotenuse, as defined in Geometry, Post 1.

Measurements of Hypotenuse segments are considered 
irrational numbers. They are also all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers, like the number 3.75 at the top of page.

Irrational numbers may also be dealt with via non-terminating continued fractions. They can be represented in precisely one way as an infinite continued fraction. See following links:

https://en.wikipedia.org/wiki/Line_segment?wprov=sfla1.

https://en.wikipedia.org/wiki/Continued_fraction?wprov=sfla1.

Triangle C can be any right triangle, and is known as the baseline triangle, from which A & B are derived, as stated in Geometry, Post 1, "Three Dissimilar Right Triangles".

Refer to the above post for description, and construction of triangles A and B.

PROPOSITION: The decimal fractional part of the dimension on the greater line segment "SA1" on hypotenuse of A is equal to the decimal fractional part of the dimension on the lesser line segment "SB2" of the hypotenuse of B, and the greater line segment "SC1" on the hypotenuse of C.

PROPOSITION (continued)
The decimal, fractional part of the dimension on the lesser line segment "SA2" on hypotenuse of A is equal to the decimal fractional part of the dimension on the greater line segment "SB1" of the hypotenuse of B, and the lesser line segment "SC2" on the hypotenuse of C. See Item 15.

1. For baseline triangle C, Pythagorean triangle 28, 45, 53 is chosen. See Items 12 thru 14 for calculations of segments SC1, SC2.

2. Segments SA1, SA2 on Hyp of A
83.20754717 Greater Seg SA1.
06.79245283 Lesser Seg SA2.
- - - - - - - -.- - - - -
90.00000000 = (2×45)*
See Items 10 and11 for segment calculations, accurate to 8 decimal places.

* Post 1, "Three Dissimilar Right Triangles" requires Hyp of A to be twice the long leg (45) of C. See NOTE above.

3. Segments SB1, SB2 on Hyp of B
42.79245283 Greater Seg SB1.
13.20754717 Lesser Seg SB2.
- - - -.- - - - -.-.- -
56.00000000 = (2×28)*
See Items 8 and 9 for segment calculations, accurate to 8 decimal places.

* Post 1, "Three Dissimilar Right Triangles" requires Hyp of B to be twice the short leg (28) of C. See NOTE above.

4. Let "d" = Altitude to Hyp in C. d = 28x45/53 = 23.77358491.

5. Altitude to Hyp in A and B is same as that in C*.

* Post 1, "Three Dissimilar Right Triangles" requires altitude to hypotenuse in A & B to be identical to that in C. See NOTE above.

6. Acute angles in A & B are 1/2 the acute angles in baseline triangle C.

7. For instance; Triangle A has 1/2 the lesser acute angle in C. Triangle B has 1/2 the greater acute angle in C.

8. Use formula below for greater Seg SB1 on Hyp of B; a = short leg of C; b = long leg of C; c = hypotenuse of C; d = altitude to hypotenuse of C.

SB1= b x d / (c - a)
SB1=45 x 23.77358491/(53 - 28) =
42.79245283. See Item 9 for Seg SB2. See Item 3.*

9. SB2 on Hyp of B = 56.00000000 - 42.79245283 = 13.20754717. See Item 3.*

* Post 1, "Three Dissimilar Right Triangles" requires Hyp of B to be twice the short leg (28) of C. See NOTE above.

10. Use formula below for greater Seg SA1 on Hyp of A. See Item 2.

SA1= a x d / (c - b).
SA1=28x23.77358491/ (53 - 45) = 83.20754717. See Item 11 for SA2. See Item 2.*

11. SA2, on Hyp of A = 90.0000000 - 83.20754717 = 6.79245283. See Item 2.*

* Post 1, "Three Dissimilar Right Triangles" requires Hyp of A to be twice the long leg (45) of C. See NOTE above.

12. For greater Seg SC1 on Hyp of C, use formula below; where b = long leg of C; c = hypotenuse. See Item 1.

SC1 = b^2 / c
SC1 = 45^2 / 53 = 38.20754717.

13. For lesser Seg SC2 on Hyp of C, use formula below, where a = short leg of C; c = hypotenuce. See Item 1.

SC2 = a^2 / c
SC2 = 28^2 / 53 = 14.79245283.

14. For sum of SC1, SC2. See below;

38.20754717 = SC1.
14.79245283 = SC2.
- - - - - - - - - - - -
53.00000000. See Item 1.

15. The following is a comprehensive review of the above, comparing segment dimensions and their decimal fractional parts. See Proposition at top.

83.20754717 = SA1
6.79245283 = SA2
- - - - - - - - - - - -
90.00000000 = 2x45 = Hyp of A.
                                            
42.79245283 = SB1
13.20754717 = SB2
- - - - - - - - - - - -
56.00000000 = 2x28 = Hyp of B.

38.20754717 SC1
14.79245283 SC2
- - - - - - - - - - - -
53.00000000 = Hyp of C.

15A. The decimal fraction .20754717 of Seg SA1 equals the decimal fraction of SB2 & SC1. See Proposition.

15B.The decimal fraction .79245283 of SA2 equals the decimal fraction of SB1 & SC2. See Proposition.

15C. The sum of the decimal fractions .79245283 and .20754717 equals 1.00000000.

15D. The square root of the products of the greater and lesser segments, SA1, SA2 and SB1, SB2, on each of the hypotenuses of triangles A & B, is equal to the square root of the product of the greater and lesser segments SC1, SC2 on the hypotenuse of triangle C.

15E.The above quantity is known as the Geometric Mean between the 2 segments on the hypotenuse of a right triangle. Also known as the altitude to the hypotenuse, as described in the following link;

https://en.wikipedia.org/wiki/Geometric_mean_theorem?wprov=sfla1.

15F.The altitude to the hypotenuse of any right triangle is equal the product of the 2 legs divided by the hypotenuse. For example; in baseline triangle C (28, 45, 53); Alt to Hyp = 28 x 45 / 53 =
23.77358491.

NOTE: The above quantity was used in Items 8 &10 to calculate the greater segments SA1 & SB1.

15G. The square of the altitude to the hypotenuse of any right triangle is equal to the product of the 2 segments on the hypotenuse. For triangles A, B & C, that product is 565.1833393.

15H. The square root of 565.1833393
is 23.77358491, the altitude to the hypotenuse on triangles A, B & C. See Items 4, 5 and 15F.