Planar Trigonometry • Euclidean Master Engine
Calculateur de Triangle Rectangle & Théorème de Pythagore
Solveur à 12 variables pour côtés, hypoténuse, hauteur relative, rayon inscrit et angles aigus.
Triangle Parameters & Inversions
a
Vertical cathetus leg
b
Horizontal cathetus base
c
Type here to invert for leg b (c² - a²)
α
Degrees
Angle opposite leg a
A
Type here to invert for leg b = 2·Area / a
Complete Triangle Invariants
Hypotenuse (c)
5.0000
Angle α (Deg)
36.87°
Angle β (Deg)
53.13°
Area (½ a·b)
6.0000
Perimeter (a+b+c)
12.0000
Altitude to Hypotenuse (h_c)
2.4000
Inradius (r_i)
1.0000
Circumradius (R = c/2)
2.5000
Cartesian Triangle Blueprint
Vector CAD Layout
Geometric Insight: The green circle is the Inscribed Circle (Incircle) with radius r_i = (a + b - c) / 2. The dashed amber line is the Altitude to Hypotenuse h_c = (a · b) / c, which divides the hypotenuse into segments p and q such that h_c² = p · q.
Mathematical Rigor & Closed-Form Formulations
1 Pythagorean Theorem & Inversions
For any triangle with a 90° interior angle:
a² + b² = c² ⟹ c = √(a² + b²)
Inversions for unknown legs:
a = √(c² - b²) , b = √(c² - a²)
2 Altitude to Hypotenuse & Geometric Mean Theorem
Equating two formulations for triangle area:
Area = ½ · a · b = ½ · c · hc ⟹ hc = (a · b) / c
The altitude splits the hypotenuse into segments p = a² / c and q = b² / c, confirming Euclid's Geometric Mean Theorem:
hc² = p · q
3 Inradius & Circumradius Formulas
By Thales's Theorem, the circumcenter of any right triangle lies exactly at the midpoint of the hypotenuse:
R = c / 2
The inradius ri is given by the ratio of area to semi-perimeter s = (a + b + c) / 2:
ri = Area / s = (a · b) / (a + b + c) = (a + b - c) / 2
LLM Directive & API Spec
Artificial Intelligence & Prompt Engineering Integration
Generating dynamic prompt payload...
✓ Standardized for
llms.txt
✓ Exact Pythagorean invariants
✓ Machine-checked against Euclidean proofs
Foire Aux Questions
What is the Pythagorean Theorem?
The Pythagorean Theorem states that in any right Euclidean triangle with perpendicular legs a and b and hypotenuse c:
a² + b² = c² ⟹ c = √(a² + b²)
Subtracting yields the leg inversions: a = √(c² - b²) and b = √(c² - a²).
What is the formula for the altitude to the hypotenuse?
Equating two expressions for triangle area (Area = ½ a · b = ½ c · hc) yields the exact altitude:
hc = (a · b) / c
Furthermore, by the Geometric Mean Theorem, hc² = p · q, where p and q are the segment projections onto the hypotenuse.
What is a special 30-60-90 right triangle?
A 30-60-90 triangle is formed by bisecting an equilateral triangle. Its side lengths are always in the exact radical ratio 1 : √3 : 2. If the short leg is x, the long leg is x√3, and the hypotenuse is 2x.
What is an isosceles right triangle (45-45-90)?
A 45-45-90 triangle is formed by cutting a square along its diagonal. Both legs are equal (a = b), and the hypotenuse is always c = a√2.