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### Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624

# Prove that the midpoint of the line segment between two constructible points is a constructible point. [Hint: Adapt the hint to Exercise ${\mathbf{17}}$ .]

The midpoint of the segment between two constructible points is a constructible point.

See the step by step solution

## Step 1: Conceptual Introduction

Drawing precise lines, line segments, shapes, circles, and other figures with a ruler, compass, or protractor is known as geometric construction.

## Step 2: Properties of constructible.

1. A point $\left(r,s\right)$ is constructible iff $r$ and $s$ are constructible.
2. If $a,b,c$ with $c\ne 0$ are constructible then $a+b,a-b$ and $a/c$ are constructible.

## Step 3: Proving that the midpoint of the line segment between two constructible point is also constructible.

The proving part:

Let $ℚ\left[x\right]$ , $\text{\hspace{0.17em}}\left(\text{\hspace{0.17em}}{a}_{1},{a}_{2}\right)$and $\text{\hspace{0.17em}}\left(\text{\hspace{0.17em}}{b}_{1},{b}_{2}\right)$ are constructible points.

Then from $1$ , each $\text{\hspace{0.17em}}{a}_{1},{a}_{2},{b}_{1},{b}_{2}$ are constructible.

Form point$2$ in step 1st, we infer that $\frac{{a}_{1}+{b}_{1}}{2}$ and $\frac{{a}_{2}+{b}_{2}}{2}$ are constructible numbers.

Then by reverse of point $1$ in step 1st, the points $\left(\frac{{a}_{1}+{b}_{1}}{2},\text{\hspace{0.17em}}\frac{{a}_{2}+{b}_{2}}{2}\right)$ is constructible.

Hence, the midpoint of the line segment between two constructible points is a constructible point.