Geometric Bites

The Pythagorean Hexagon — Proof and Definition © Tiago Hands — 18 October 2025 (UTC +8) Definition and Construction Six points A, B, C, D, E and F form a hexagon whose sides are all equal in length. Point G lies at the centre of a circle with radius AB = x > 0. Points F, B and D lie on the circumference of this circle, forming a right-angled triangle at G where angle FGB = 90°. Quadrilateral ABGF forms a square. Quadrilaterals FGDE and BCDG form rhombuses that share the same side length x. The internal angles satisfy α + β = 270°. Each figure has a centre: H for square ABGF, I for rhombus BCDG, and J for rhombus FGDE. When these three figures are joined edge-to-edge around the circle, their areas obey a Pythagorean relation. Areas of the Component Figures Let AB = x. Square ABGF: A 1 = x 2 Rhombus FGDE: A 2 = 2x 2 cos(β/2)sin(β/2) = x 2 sin(β) Rhombus BCDG: A 3 = 2x 2 cos(α/2)sin(α/2) = x 2 sin(α) Algebraic and Trigonometric Proof We aim to show A...

The Pythagram Defined Contributors over a 2.5 Year Period: Tiago Hands ( Final Construction ), Carlos Luna-Mota ( Egyptian Triangles ), Andrzej Kukla ( The Rhombus with Area of 3 ) 06 October 2025 Abstract The Pythagram is a planar geometric structure derived from the 3 : 4 : 5 right triangle and the orthographic projection of a cube. Its construction through defined Cartesian coordinates reveals six interrelated Pythagorean sub-figures that encode proportional symmetry across multiple dimensions. This document formalises the coordinates, connections, and geometric properties of the Pythagram as a reproducible mathematical entity. Coordinate Data Group 1 — Central Core A(16, 18), B(14, 18), C(12, 16), D(12, 14), E(14, 12), F(16, 12), G(18, 14), H(18, 16), I(15, 15), W(15, 17.5), X(12.5, 15), Y(15, 12.5), Z(17.5, 15) Group 2 — Inner Square L(15, 20), M(10, 20), N(10, 15), O(10, 10), P(15, 10), Q(20, 10), R(20...

Below I demonstrate how to derive tan(π/8) , sin(π/8) and cos(π/8) from scratch. These workings are a bit similar to the workings for tan((3π)/8) , sin((3π)/8) and cos((3π)/8) which can be found here : https://geometricbites.blogspot.com/2021/08/how-to-derive-tan38-sin38-and-cos38.html The edge with length R √(2- √(2)) was actually derived in the tan((3π)/8), sin((3π)/8) and cos((3π)/8) post, so doesn't need to be found again. Part 1  Part 2  Part 3  Part 4

Below I demonstrate how to derive tan((3π)/8) , sin((3π)/8) and cos((3π)/8) from scratch. I use geometry and also algebra . Part 1  Part 2 Part 3 Part 4 Part 5  Part 6  If you'd like quick updates every time I post, follow me on Twitter at: https://twitter.com/tiago_hands .

Below I demonstrate how to find the line that passes through two intersecting circles. Part 1 Part 2 Part 3   There is a free interactive graph that shows why this is correct. https://www.desmos.com/calculator/z9rh2ri9bz