Pythagorean Triples + Sagerian Triangles

Generate triples (c ≤ 10,000), click one to preview the triangle, then visualize Sagerian Triangles.

Ready.

Ceiling for hypotenuse (c)

Max is 10,000 for performance.

0 triples

#	a	b	c	Angle 1	Angle 2	Primitive
Generate triples to see results here.

Tip: click a row to update the preview. Angles shown are the two acute angles (right angle omitted).

Selected Triple

a—

b—

c—

Angles—

Area—

Perimeter—

Triangle Preview

Original Rotated copy Overlap Outlined triangles

Sagerian Result

Press Sagerian Triangles to compute and visualize the Sagerian construction for the selected triple.

Advanced

Show Berggren tree addresses

Selected (normalized primitive)—

Selected address—

Selected scale—

Sagerian (normalized primitive)—

Sagerian address—

Sagerian scale—

Addresses are computed in the Berggren tree (matrices A/B/C) for the primitive triple (odd leg, even leg, hypotenuse).

What are Sagerian Triangles?

Start with a Pythagorean triple (a, b, c) where a is the smaller leg, b is the larger leg, and c is the hypotenuse.

Compute the Sagerian scale factor: M = b · c, and scale the triangle to (aM, bM, cM).
Copy that scaled triangle and rotate the copy by 90° (the layout creates an overlap region).
The resulting figure contains three non-overlapping triangles similar to the original.
Those three triangles are exactly k₁, k₂, and k₃ times larger than the original triangle.

The three multipliers can be computed directly from (a,b,c):

k₁ = c(b − a)

k₂ = ab

k₃ = a² − ab + b²

These multipliers form a new Pythagorean triple: k₁² + k₂² = k₃².

In the app: select a triple to view the basic triangle. Press Sagerian Triangles to reveal the rotated copy, show the overlap, and compute/verify (k₁,k₂,k₃).