Counting Triangles in Triangles
Jim Propp, Adam Propp-Gubin
TL;DR
This paper presents formulas for counting triangles formed by the sides and cevians of a triangle, enabling proofs of existing claims and discovery of new results in geometric combinatorics.
Contribution
It introduces a novel formula for counting triangles in a triangle with cevians, including special cases where cevians divide sides equally.
Findings
Validated formulas for counting triangles in various configurations.
Proved previously unverified claims from OEIS and popular videos.
Derived new formulas for equal-length segment divisions on sides.
Abstract
We give a formula for counting the triangles in a picture consisting of the three sides of a triangle and some cevians. This lets us prove statements that are claimed without proof in the Online Encyclopedia of Integer Sequences and some popular YouTube videos, and also prove some new results. We also give formulas that apply when the cevians cut each side into equal-length pieces.
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Taxonomy
TopicsHistory and Theory of Mathematics · Mathematics and Applications