sidef-scripts/Math/bisected_hypotenuse.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 06 July 2018
# https://github.com/trizen

# Given two catheti lengths (A+B and C) of a right triangle and assuming a parallel line,
# relative to side C, that divides the triangle, find the values of X and Y, satisfying:
#
#   1. (X+Y)^2 = (A+B)^2 + C^2
#   2. X/Y = A/B
#
#               C
#   ----================------- L1
#        \             |
#     X   \            |     A
#          \           |
#   --------------------------- L2
#            \         |
#     Y       \        |     B
#              \       |
#               \      |
#                \     |
#                 \    |
#                  \   |
#                   \  |
#                    \ |
#                     \|

# Because the lines L1 and L2 are parallel to each other, we have the identity:
#   A/B = X/Y

# See also:
#   https://www.youtube.com/watch?v=ffvojZONF_A

func approximate_solution(A, B, C) {    # approximation in positive integers

    var h = sqrt((A+B)**2 + C**2)
    var r = (A > B ? (A / B) : (B / A))

    var Y = bsearch_le(0, h, {|n|
        n / (h-n) <=> r
    })

    var X = h-Y
    (X, Y) = (Y, X) if (A > B)
    return [X,Y]
}

func exact_solution(A, B, C) {          # exact solution in complex numbers

    var X = (1i * A * sqrt(A**2 + 2*A*B + B**2 + C**2))/sqrt(-((A+B)**2))
    var Y = (1i * B * sqrt(A**2 + 2*A*B + B**2 + C**2))/sqrt(-((A+B)**2))

    return [X, Y]
}

func exact_solution_real(A, B, C) {     # exact solution in real numbers

    var X = (A * sqrt((A+B)**2 + C**2))/abs(A+B)
    var Y = (B * sqrt((A+B)**2 + C**2))/abs(A+B)

    return [X, Y]
}

say approximate_solution(49, 161-49, 240)    #=> [88, 201]
say approximate_solution(161-29, 29, 240)    #=> [236, 53]
>''
say exact_solution(49, 161-49, 240)          #=> [87.9565217391304347826086956521739130434782608696, 201.04347826086956521739130434782608695652173913]
say exact_solution(161-29, 29, 240)          #=> [236.944099378881987577639751552795031055900621118, 52.055900621118012422360248447204968944099378882]
>''
say exact_solution_real(49, 161-49, 240)     #=> [87.9565217391304347826086956521739130434782608696, 201.04347826086956521739130434782608695652173913]
say exact_solution_real(161-29, 29, 240)     #=> [236.944099378881987577639751552795031055900621118, 52.055900621118012422360248447204968944099378882]