Apply a homothety (geometric scaling) to any triangle. Choose scale factor k and scaling center (centroid, origin, or custom point). Visualize original vs scaled triangle with interactive canvas. Perfect for learning similarity transformations, dilation effects, and coordinate geometry.
A homothety (or dilation) is a transformation that maps every point P to a point P' such that P' = C + k · (P − C), where C is the fixed scaling center and k is the scale factor (non‑zero real number). This transformation preserves angles and parallelism, mapping any triangle onto a similar triangle with side lengths multiplied by |k|. For k > 0, the scaled triangle lies on the same side of C; for k negative, it is rotated 180° about C (central symmetry combined with scaling). The ratio of areas equals k².
If C = (cₓ, cᵧ) and scale factor k, then:
x' = cₓ + k·(x − cₓ) , y' = cᵧ + k·(y − cᵧ)
This linear transformation is a central dilation, fundamental in Euclidean geometry and computer graphics.
The concept of homothety appears in the works of Euclid (Elements, Book VI) through the theory of similar figures. The term "homothety" was coined by French mathematician Michel Chasles in the 19th century. Dilation is the foundation of similarity transformations, used extensively in Thales' theorem, intercept theorems, and geometric constructions. In modern mathematics, scaling about a center is a special case of an affine transformation and is crucial in fractal geometry (self‑similarity), computer‑aided design, and cartography (map scaling).
For a triangle with vertices A, B, C and scaling center C₀ = (cₓ, cᵧ), the dilated vertices A', B', C' are computed using vector scaling: A' = C₀ + k·(A − C₀). This preserves the ratio of distances from C₀: |C₀A'| / |C₀A| = |k|. Our calculator solves this directly for any user‑defined center (centroid, origin, or custom). The centroid G is computed as ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3). The area is calculated via the shoelace formula; the area ratio always matches k² (theoretical check).
For k negative, the transformed triangle is rotated 180° about the center, representing a point reflection combined with magnitude scaling. This creates a centrally symmetric dilation widely used in advanced geometry proofs.
Click any preset example to instantly see the effect. For instance, scaling a right triangle with k=1.5 about its centroid enlarges the triangle while keeping the centroid fixed. Choosing k = -1 rotates the triangle 180° around the center, producing a congruent triangle but reversed orientation. Our canvas displays both original (gray) and scaled (blue) triangles, plus the scaling center (green dot).
| Triangle Type (Original) | Scale Factor k | Scaling Center | Effect on Position / Size | Area Ratio |
|---|---|---|---|---|
| Right (3-4-5) | 2.0 | Centroid | Doubles side lengths, centroid fixed | 4.00 |
| Equilateral | 0.5 | Origin | Shrinks toward origin | 0.25 |
| Acute scalene | -1.5 | Custom (2,2) | Enlarges and reflects across custom center | 2.25 |
| Obtuse | 1.0 | Any | Identical triangle (k=1) | 1.00 |
An architect designs a triangular atrium with vertices A(0,0), B(10,0), C(5,8). To fit a 1:50 scale model, a dilation about the centroid with factor k=0.02 is applied. The calculator instantly yields scaled coordinates, allowing precise CAD adjustments. Moreover, when creating a mirrored annex across the centroid, a negative scale factor k = -1 produces a rotationally symmetric design. This geometric scaling tool streamlines iterative design and ratio verification.