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 fractional, or decimal part of the positive real number 3. The decimal fractional part, .75 = the fraction 3/4.
For sources, see links below;
Dimensions with fractional, or decimal parts, invariably occur on the two segments of the hypotenuse of a right triangle, caused by the altitude to the hypotenuse.
In the link below, see segments "p" and "q".
NOTE: Segments "p" and "q" are used in this post to help visualize the graphical association with actual dimensions calculated for the identical type of segments on the hypotenuse.
This post is about the obscure, and frequent occurrence of repeating patterns of identical decimal fractions, of actual dimensions, of segments on hypotenuses of 3 right triangles, A, B, & C, as defined in Geometry, Post 1.
Please refer to Post 1 for description, and construction of triangles A and B. Triangle C is a known baseline right triangle, from which A & B are derived.
1. For C, Pythagorean triangle 28, 45, 53 is selected.
2. Hyp on triangle C = 2 x 45 = 90.
83.20754717 Greater Seg SC1.
6.79245283 Lesser Seg SC2.
- - - - - - - -.- - - - -
See Items 10 and11 for segment calculations.
* Post 1, "Three Dissimilar Right Triangles" requires Hyp of C to be twice the long leg of C. See NOTE aboveve.
3. Hyp on triangle B = 2 x 28 = 56. 42.79245283 Greater Seg SB1.
13.20754717 Lesser Seg SB2.
- - - -.- - - - -.-.- -
See Items 8 and 9 for segment calculations.
* Post 1, "Three Dissimilar Right Triangles" requires Hyp of B to be twice the short leg of C. See NOTE above.
4. Altitude to Hyp in C=28x45/53 =
5. Altitude to Hyp in A and B is same as 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 triangles A & B are 1/2 the acute angles in baseline triangle C.
6a.Triangle A has 1/2 the lesser acute angle in C.
6b.Triangle B has 1/2 the greater acute angle in C. See NOTE above.
7. PROPOSITION: The decimal fractional part of the dimension on the greater segment on hypotenuse of A is equal to the decimal fractional part of the dimension on the lesser segment of the hypotenuse of B, and the greater segment on the hypotenuse of C
8. Use formula SB1=b x d / (c - a)
for greater Seg on Hyp of B.
Triangle C is 28 45 53. a = 28, b = 45, c = 53, d = 23.77358491, Altitude to Hyp. See Items 3 and 4.
SB1=b x d / (c - a)
SB1=45 x 23.77358491/(53-28) =
42.79245283 Greater Seg on Hyp of B. Hyp=56=(2×28). See Item 3.
9. Lesser Seg SB2 on Hyp of B = 56.00000000 - 42.79245283 = 13.20754717. See Item 3.
10. For Greater Seg on Hyp of C, use formula SC1=a x d / (c - b).
a = 28, b= 45, c = 53, and d = 23.77358491. See Item 2.
SC1= a x d / (c - b) ,
SC1=28x23.77358491/ (53-45) = 83.20754717 Greater Seg on Hyp of C. Hyp = 90 (2×45). See Item 2.
11. Lesser Seg SC2, on Hyp of C = 90.0000000 - 83.20754717 = 6.79245283. See Item 2.
12. For Greater Seg on Hyp of A, use formula SA1 = b^2 / h, where b = long leg, c = hypotenuse
SA1 = 45^2 /53 = 38.20754717.
13. For Lesser Seg on Hyp of A, use formula SA2 = a^2 / h, where a = short leg. SA2 = 28^2 /53 = 14.79245283.