Axiomatic Geometry | Module II

The Fundamental Theorem of Similarity

From Dilation to Proportionality: Bridging Euclidean Ratios and Transformational Mapping.

FTS // SECTION_02

The Bridge

Last lesson, we defined a dilation as a mapping from a center.

"But why does this transformation preserve the ratio of segments across the entire plane?"

The Fundamental Theorem of Similarity (FTS) provides the logical glue connecting our mapping to Euclidean proportionality.

Theorem Statement

Given \(\triangle OAB\), let \(A' \in \vec{OA}\) and \(B' \in \vec{OB}\).

If \(\frac{OA'}{OA} = \frac{OB'}{OB} = k\), then \(A'B' = k \cdot AB\) and \(\overleftrightarrow{A'B'} \parallel \overleftrightarrow{AB}\).

The Classical Problem

Commensurable Ratios

Ancient Greek mathematicians proved similarity by dividing segments into "common measures" (rational units).

:

If \(OA'/OA = m/n\), we can divide \(OA\) into \(n\) equal parts and \(OA'\) into \(m\) of those parts.

The Crisis

What happens when the ratio \(k\) is irrational (e.g., \(\sqrt{2}\))?

The "common measure" disappears. This realization by the Pythagoreans shattered early geometry and led to the theory of real numbers.

Logic Flow

The Proof Architecture

1

Apply the Side-Splitter Theorem: If lines are parallel, they cut segments proportionally.

2

Use the Converse: If segments are cut proportionally, the line must be parallel.

3

Conclusion: The image line \(A'B'\) is parallel to \(AB\), which then forces the length \(A'B'\) to be \(k \cdot AB\) via the properties of parallelograms and similarity axioms.

Geometric Visual

"Ratio \(OA'/OA = OB'/OB = k\) implies \(A'B' \parallel AB\)"

The Power of FTS

Universal Scaling

FTS tells us that once we fix the scale of two sides of a triangle, the third side is automatically determined. There is no "wiggle room" for the base of the triangle to be any other length.

Foundation for AA

Without FTS, we could not rigorously prove that having two equal angles is enough to guarantee proportional sides. FTS is the logical precursor to all triangle similarity criteria.

NEXT: Constructing Rigorous Proofs of AA, SAS, SSS

The Fundamental Theorem of Similarity

Fundamental Theorem Similarity Slides

Fundamental Theorem Teacher Guide

Ratios Rationale Worksheet

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