Axiomatic Geometry | Module I

Defining Dilations & Homothety

Establishing the transformational foundations of similarity through rigorous axiomatic definitions.

01 // DILATION_FOUNDATIONS

AXIOM_SIMILARITY

The Scaling Challenge

In Euclidean geometry, we often rely on intuition to describe "zooming in" or "zooming out."

"How can we map a figure to a larger version of itself without resorting to a coordinate system or simple measurement?"

We need a formal mapping—a transformation—that preserves shape while changing size.

Definition 1.1

The Dilation (Homothety)

Given a point \(O\) (center) and a real number \(k > 0\) (scale factor), a dilation \(D_{O,k}\) is a transformation such that for any point \(P\):

If \(P = O\), then \(D_{O,k}(P) = O\).
If \(P \neq O\), then \(P' = D_{O,k}(P)\) is the unique point on the ray \(\vec{OP}\) such that:
\(OP' = k \cdot OP\)

Case \(k > 1\)

An Expansion. The image is further from \(O\) than the pre-image.

Case \(0 < k < 1\)

A Contraction. The image is closer to \(O\) than the pre-image.

Theorem 1.1

Dilation of Lines

A dilation \(D_{O,k}\) maps a line \(L\) to a line \(L'\) such that:

If \(L\) passes through \(O\), then \(L' = L\).
If \(L\) does not pass through \(O\), then \(L' \parallel L\).

Proof Logic

By taking points \(A\) and \(B\) on \(L\), we create \(\triangle OAB\). Their images \(A'\) and \(B'\) form \(\triangle OA'B'\). By construction, side ratios are constant, implying parallel segments.

Property 1.1

The Length Property

If \(P'\) and \(Q'\) are the images of \(P\) and \(Q\) under \(D_{O,k}\), then:

P'Q' = k · PQ

This implies that dilations preserve the ratio of distances, even if they do not preserve the distances themselves.

Seminar Discussion

Question 1

Why is it necessary to specify \(k > 0\) in our definition of homothety? What geometric meaning would a negative \(k\) carry?

Question 2

If a dilation maps line segments to parallel line segments, does the converse hold? Are all parallel segments related by a dilation?

Logic Check

"Consider the composition of two dilations with the same center \(O\) and factors \(k_1, k_2\). Prove that the result is a dilation with center \(O\) and factor \(k_1 k_2\)."

Axiomatic Foundations of Similarity

Defining Dilations and Homothety

The Fundamental Theorem of Similarity

Rigorous Proofs of AA SAS and SSS

Proportionality in Right Triangles

Power of a Point and Cyclic Quadrilaterals

Dilation Foundations Slides

Dilation Foundations Teacher Guide

Homothety Proportionality Worksheet

Fundamental Theorem Similarity Slides

Fundamental Theorem Teacher Guide

Ratios Rationale Worksheet

Similarity Criteria Proofs Slides

Similarity Criteria Teacher Guide

Triangle Synthesis Worksheet

Right Triangle Proportionality Slides

Geometric Mean Teacher Guide

Nested Symmetry Worksheet

Circle Similarity Slides

Circle Similarity Teacher Guide

Circle Invariants Worksheet

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