notebook/notes/geometry/index.md

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title: Geometry
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## Overview

Two sets are **congruent** if their points can be put in one-to-one correspondence in such a way that distances are preserved.

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Basic
Suppose sets $P$ and $Q$ are congruent. What does this imply the existence of?
Back: A bijection between $P$ and $Q$ that preserves distances between points.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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%%ANKI
Basic
Suppose sets $P$ and $Q$ are congruent and $f$ is the corresponding bijection. What FOL proposition follows?
Back: $\forall p_1, p_2 \in P, \lvert p_1 - p_2 \rvert = \lvert f(p_1) - f(p_2) \rvert$
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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## Bibliography

* Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).