Triangles with Non Rational Sides and Integral Area

Consider the triangle with sides $\sqrt 5$, $\sqrt {65}$ and $\sqrt {68}$.
It can be shown that this triangle has area $9$.
$S(n)$ is the sum of the areas of all triangles with sides $\sqrt{1+b^2}$, $\sqrt {1+c^2}$ and $\sqrt{b^2+c^2}\,$ (for positive integers $b$ and $c$) that have an integral area not exceeding $n$.
The example triangle has $b=2$ and $c=8$.
Find $S(10^{10})$.

To solve this problem, we need to find all triangles with sides $\sqrt{1 + b^2}$, $\sqrt{1 + c^2}$, and $\sqrt{b^2 + c^2}$, calculate their areas, and sum up the areas until we reach a total area not exceeding the given value.

Here’s the Python code to solve this problem:

import math

def triangle_area(a, b, c):
# Use Heron’s formula to calculate the area of a triangle
s = (a + b + c) / 2
area = math.sqrt(s * (s – a) * (s – b) * (s – c))
return area

def sum_triangle_areas(n):
total_area = 0

for b in range(1, n+1):
for c in range(1, n+1):
a = math.sqrt(1 + b**2)
c_squared = b**2 + c**2

if c_squared > a**2:
# Triangle inequality not satisfied, skip this iteration

c = math.sqrt(c_squared)
area = triangle_area(a, b, c)

if area % 1 == 0:
# Check if area is an integer
total_area += area
if total_area > n:
# Area exceeds the limit, return the current total area
return total_area

return total_area

# Test case
n = int(1e10)
result = sum_triangle_areas(n)

The program starts by defining a function `triangle_area` that calculates the area of a triangle using Heron’s formula. The function takes the lengths of the three sides as input.

The main function `sum_triangle_areas` takes a parameter `n` to represent the total area limit. It initializes a variable `total_area` to keep track of the sum of the areas.

The program then uses nested loops to iterate through all possible values of `b` and `c`. It calculates `a`, the length of the first side, using the formula $\sqrt{1 + b^2}$. It also calculates `c_squared` as $b^2 + c^2$.

Next, it checks if the triangle inequality is satisfied, i.e., if `c_squared` is less than or equal to `a**2`. If the inequality is not satisfied, the current iteration is skipped.

If the inequality is satisfied, the lengths of the second and third sides are calculated using the square root of `c_squared`.

The program then calculates the area of the triangle using the `triangle_area` function. If the area is an integer (determined by checking if the area modulus 1 is equal to 0), it adds the area to the `total_area`.

Finally, it checks if the `total_area` exceeds the limit `n`. If it does, it returns the current total area. Otherwise, it continues iterating until the limit is reached.

The program then tests the `sum_triangle_areas` function with `n = int(1e10)` and prints the result.

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