2+ mon, 1+ day ago   (58+ words) Here are four theorems that generalize the Pythagorean theorem. Follow the links for more details regarding each equation. 1. Theorem by Apollonius for general triangles. 2. Edsgar Dijkstra's extension of the Pythagorean theorem for general triangles. 3. A generalization of the Pythagorean theorem…...

2+ mon, 6+ day ago   (195+ words) The area of a triangle can be computed directly from the lengths of its sides via Heron's formula. Here's is the semiperimeter,'s = (a +'b +'c)/2. Is there an analogous formula for spherical triangles? It's not obvious there should be, but…...

2+ mon, 1+ week ago   (216+ words) A tetrahedron has four triangular faces. Suppose three of those faces come together like the corner of a cube, each perpendicular to the other. Let A1, A2, and A3 be the areas of the three triangles that meet at this corner and let…...

2+ mon, 2+ week ago   (123+ words) Apollonius of Perga (c. 262 BC " c. 190 BC) discovered a theorem that generalizes the Pythagorean theorem but isn't nearly as well known. Let ABC be a general triangle, and let D be the midpoint of the segment AB. Let'a be the length of…...