Euclid’s Elements, Book I: equal areas between parallels

Euclid’s theory of area (Elements I.35-41) rests on two facts: parallelograms on the same base and between the same parallels are equal in area (I.35), and so are triangles (I.37), each triangle being half of the parallelogram (I.41). Sliding the top side along its parallel line changes every shape but, as the shoelace formula in polygon_area() confirms, never the area.

import matplotlib.pyplot as plt
import numpy as np

from mathematicskit.geometry import polygon_area, polygon_centroid

Parallelograms on one base between two parallels (I.35)

The base runs from (0, 0) to (4, 0); the opposite side lies on the parallel line y = 2, shifted sideways by different amounts.

base, height = 4.0, 2.0
shifts = [0.0, 2.5, 6.0]
parallelograms = [np.array([[0, 0], [base, 0], [base + s, height], [s, height]]) for s in shifts]
for s, quad in zip(shifts, parallelograms):
    cx, cy = polygon_centroid(quad)
    print(f"shift {s:3.1f}: area {polygon_area(quad):.6f}, centroid ({cx:.2f}, {cy:.2f})")
print(f"base x height = {base * height:.6f}")

shift 0.0: area 8.000000, centroid (2.00, 1.00)
shift 2.5: area 8.000000, centroid (3.25, 1.00)
shift 6.0: area 8.000000, centroid (5.00, 1.00)
base x height = 8.000000

Triangles on the same base, half the parallelogram (I.37, I.41)

apexes = [0.0, 3.0, 7.0]
triangles = [np.array([[0, 0], [base, 0], [a, height]]) for a in apexes]
for a, tri in zip(apexes, triangles):
    print(f"apex at x = {a:3.1f}: area {polygon_area(tri):.6f} (half of {base * height:.1f})")

apex at x = 0.0: area 4.000000 (half of 8.0)
apex at x = 3.0: area 4.000000 (half of 8.0)
apex at x = 7.0: area 4.000000 (half of 8.0)

The figures

Every centroid lies on the midline y = h/2 for the parallelograms and y = h/3 for the triangles: the shapes shear, their areas do not change.

fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(9, 4.5), sharex=True)
colors = ["tab:blue", "tab:orange", "tab:green"]
for ax, shapes, title in [(ax1, parallelograms, "I.35: parallelograms"), (ax2, triangles, "I.37: triangles")]:
    ax.axhline(0, color="k", lw=0.8)
    ax.axhline(height, color="k", lw=0.8, ls="--")
    for shape, color in zip(shapes, colors):
        ax.fill(*shape.T, color=color, alpha=0.3, ec=color, lw=1.5)
        ax.plot(*polygon_centroid(shape), "o", color=color)
        ax.text(*polygon_centroid(shape), f"  {polygon_area(shape):.1f}", color=color, va="center")
    ax.set_aspect("equal")
    ax.set_title(f"{title} on the same base between the same parallels")
plt.show()