TheoremProved

Scaling of Distances and Areas Under Homothety

Statement

Under a homothety $V_{(I, k)}$ with center $I$ and ratio $k \neq 0$ , for any two points $A, B$ with images $A' = V_{(I,k)}(A)$ and $B' = V_{(I,k)}(B)$ , we have $\overrightarrow{B'A'} = k\,\overrightarrow{BA}$ and hence $A'B' = |k|\cdot AB$ . Consequently, the area of any triangle $ABC$ scales by the square of the ratio: $[A'B'C'] = k^2 \cdot [ABC]$ .

Why is it true?

Because every point is pushed away from (or pulled toward) the center $I$ by the same factor $k$ , the triangle formed by $I$ and any two points $A, B$ is scaled uniformly in both radial sides, so by Thales' theorem the third side $A'B'$ stays parallel to $AB$ and scales by $|k|$ . Area is two-dimensional (base times height), and since both base and height are multiplied by $|k|$ , their product is multiplied by $|k| \cdot |k| = k^2$ .

Proof sketch

By the definition of homothety, $\overrightarrow{IA'} = k\,\overrightarrow{IA}$ and $\overrightarrow{IB'} = k\,\overrightarrow{IB}$ . Subtracting the second vector equation from the first yields $\overrightarrow{IA'} - \overrightarrow{IB'} = k(\overrightarrow{IA} - \overrightarrow{IB})$ .

Using the head-to-tail rule $\overrightarrow{IA} - \overrightarrow{IB} = \overrightarrow{BA}$ on both sides, we obtain $\overrightarrow{B'A'} = k\,\overrightarrow{BA}$ . Taking lengths of both vectors immediately gives $A'B' = |\overrightarrow{B'A'}| = |k| \cdot |\overrightarrow{BA}| = |k| \cdot AB$ .

Now consider any triangle $ABC$ . Its altitude $h$ from $C$ to line $AB$ is the distance between $C$ and its orthogonal projection $H$ on $AB$ . Since homothety preserves parallelism and angles, it preserves perpendicularity, so $H' = V_{(I,k)}(H)$ is the foot of the altitude of $A'B'C'$ . Thus both the base $A'B' = |k|\cdot AB$ and the altitude $C'H' = |k|\cdot CH$ scale by $|k|$ , giving $[A'B'C'] = \tfrac{1}{2} A'B' \cdot C'H' = |k|^2 \cdot \tfrac{1}{2} AB \cdot CH = k^2 \cdot [ABC]$ .

Topics that use this theorem

Geometric transformations

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

H. S. M. Coxeter (1969). Introduction to Geometry
Wikipedia contributors (2026). Geometric transformation — Wikipedia