Observation of Side-Length Ratios in Equiangular Triangles

This GeoGebra example demonstrates that when two triangles are equiangular (their corresponding angles are equal), the ratio of lengths of corresponding sides remains invariant across different placements and constructions. The key ideas include equiangular geometry, similar triangles, and the proportionality of corresponding sides derived from angle correspondence. Construction highlights: GeoGebra Point is used to create defining vertices (A, B, C and their transformed points such as A′, B′, D′); Rotate performs the geometric rotation needed to align corresponding angles; Angle marks and enforces the angle conditions (α, β, γ, δ, ε, etc.); Line and Intersect build supporting lines and compute intersection points that determine the new vertex positions; Segment provides measurable side lengths used to form the side-length ratios; Circle supports the rotation/angle-guided placement of points and constraints. Suitable for secondary-level instruction and exploratory learning, especially for teaching similarity, angle-based constructions, and the “unchanged correspondence ratio” phenomenon in geometry.