Step 1: Initial Setup & The Definition of a Mid-point

Consider a line segment $AB$. By geometric definition, a point $C$ is considered the mid-point of the line segment $AB$ if it lies on the segment between $A$ and $B$, and divides it into two equal parts. Mathematically, this is expressed as:

$AC = BC$

Since point $C$ lies on the line segment $AB$, the sum of the parts equals the whole [Per Euclid's Axiom 4: Things which coincide with one another are equal to one another, and Axiom 5: The whole is greater than the part]. Therefore:

$AC + BC = AB$

Substituting $BC$ with $AC$ yields:

$AC + AC = AB \implies 2AC = AB \implies AC = \frac{1}{2}AB \quad \text{--- (Equation 1)}$

Step 2: Formulating the Hypothesis for Proof by Contradiction

To prove that a line segment has one and only one mid-point, we will utilize a proof by contradiction. We begin by assuming the opposite of our desired conclusion.

Assumption: Let us assume that the line segment $AB$ has two distinct mid-points, namely $C$ and $D$.

Step 3: Applying Euclidean Axioms to the Assumption

If $D$ is also a mid-point of the line segment $AB$, it must satisfy the exact same geometric conditions as point $C$. Therefore, $D$ divides $AB$ into two equal parts ($AD = DB$). Following the identical algebraic derivation from Step 1, we obtain:

$AD = \frac{1}{2}AB \quad \text{--- (Equation 2)}$

Now, we analyze Equation 1 and Equation 2:

• From Equation 1: $AC = \frac{1}{2}AB$
• From Equation 2: $AD = \frac{1}{2}AB$

We apply Euclid’s First Axiom: "Things which are equal to the same thing are equal to one another."

Since both $AC$ and $AD$ are equal to $\frac{1}{2}AB$, they must be equal to each other:

$AC = AD$

Step 4: Geometric Interpretation and Resolution of the Contradiction

We have established that the distance from $A$ to $C$ is exactly equal to the distance from $A$ to $D$ along the same line segment $AB$.

Because points $C$ and $D$ both lie on the line segment $AB$ and are measured from the same origin point $A$ in the same direction, the equation $AC = AD$ implies that there is zero distance between point $C$ and point $D$.

By Euclid’s Fourth Axiom: "Things which coincide with one another are equal to one another." Conversely, since the lengths are equal and share the same initial point and direction, the endpoints must coincide. Therefore, point $C$ and point $D$ are not distinct; they are the exact same point.

This directly contradicts our initial assumption that $C$ and $D$ are two distinct mid-points. Because the assumption leads to a logical impossibility, the assumption must be false.

Final Solution: By proving that any assumed second mid-point must perfectly coincide with the first, we have rigorously demonstrated that every line segment possesses one, and only one, mid-point.

Step 1: Primary Definition of a Square

In Euclidean geometry, a square is defined as a regular quadrilateral. Specifically, it is a closed, two-dimensional flat figure bounded by four straight line segments where:

• All four sides are of equal length ($AB = BC = CD = DA$).
• All four interior angles are right angles ($\angle A = \angle B = \angle C = \angle D = 90^\circ$).

[Per Euclidean classification, a square can also be defined as a rectangle with two adjacent equal sides, or a rhombus with a right interior angle.]

Step 2: Identification of Prerequisite Terms

Yes, to rigorously define a "square," several foundational geometric terms must be defined first to avoid circular reasoning. The definition of a square relies on the concepts of a quadrilateral, line segment, angle, and right angle. Furthermore, these terms rely on the undefined primitive terms of Euclidean geometry: point, line, and plane.

Step 3: Definitions of Prerequisite Terms

To construct the definition of a square from the ground up, the following sequence of definitions is required:

• Point: An undefined primitive term in modern geometry. [Per Euclid's Elements, Book I, Definition 1: "A point is that which has no part."] It represents an exact location in space with zero dimensions.
• Line: An undefined primitive term representing a straight, continuous arrangement of points extending infinitely in two directions with no breadth.
• Line Segment: A bounded portion of a line consisting of two distinct endpoints and all the points on the line strictly between them.
• Angle: A geometric figure formed by two rays (or line segments) that share a common endpoint, known as the vertex.
• Right Angle: When a straight line stands on another straight line making the adjacent angles equal to one another, each of the equal angles is a right angle ($90^\circ$).
• Polygon: A closed two-dimensional figure formed by a finite number of straight line segments connected end-to-end.
• Quadrilateral: A specific type of polygon bounded by exactly four line segments (sides).

Step 4: Geometric Visualization of a Square

The following diagram illustrates a square $ABCD$, demonstrating the equality of its four sides and the $90^\circ$ measure of its four interior angles.

Step 5: Logical Synthesis

By defining the primitive and intermediate terms, we establish a rigorous axiomatic foundation. A square cannot exist without the concept of a quadrilateral, which cannot exist without line segments, which in turn rely on points and lines. The condition of "equal sides" requires the concept of geometric congruence, and "right angles" requires the definition of angular measure.

Final Solution: A square is defined as a quadrilateral with all four sides equal and all four angles as right angles ($90^\circ$). Yes, other terms must be defined first. The prerequisite terms are "quadrilateral," "line segment," "angle," "right angle," "point," and "line," which are defined sequentially from undefined primitive concepts to complex polygons to ensure logical consistency in Euclidean geometry.