Foundations

A point is a location with no size—no length, width, or height. It's pure position.

This is where abstraction begins. Real dots have size; mathematical points don't. Everything in geometry is built from this nothing.

A line has length but no width or thickness. It's infinitely thin.

When you draw a line with a pencil, you're making an imperfect picture of something that has no thickness at all.

Where a line segment starts and stops, those are points.

This connects D1 and D2—points and lines are related.

A straight line doesn't curve or bend—it's the shortest path between any two points on it.

Euclid's definition is poetic but vague. We intuitively know what 'straight' means.

A surface has length and width but no thickness. It's perfectly flat and thin.

Like a line but in two dimensions instead of one.

The boundaries of a surface are lines.

Connects surfaces back to lines, just as D3 connected lines to points.

A flat surface—like an infinite tabletop with no curves or bumps.

Most of Book I happens on a single plane surface.

An angle is formed when two lines meet at a point and aren't just one straight line.

The 'opening' between two lines meeting at a point.

When both sides of an angle are straight lines (not curves), it's a rectilinear angle.

Almost all angles in Book I are rectilinear.

When a line meets another and creates two equal angles on either side, those are right angles (90°). The line making them is perpendicular.

Right angles are fundamental—they're the 'standard' angle against which others are measured.

An obtuse angle is bigger than 90°—it's 'wide open.'

Between 90° and 180°.

An acute angle is smaller than 90°—it's 'sharp.'

Between 0° and 90°.

A boundary is the edge or limit of something.

General definition that applies to all figures.

A figure is a shape enclosed by one or more boundaries.

Circles, triangles, squares—all are figures.

A circle is a shape where every point on the edge is the same distance from the center.

That constant distance is the radius. This definition is crucial—circles appear in almost every construction.

The center is the special point inside from which all points on the circle are equidistant.

Continuation of D15.

A diameter is a line through the center that touches the circle on both ends. It cuts the circle in half.

The diameter is twice the radius.

A semicircle is half a circle—the region between a diameter and the arc it cuts off.

The center of a semicircle is the same as the full circle's center.

Shapes made of straight lines: triangles have 3 sides, quadrilaterals have 4, and polygons can have more.

This sets up the vocabulary for the rest of the book.

Equilateral = all 3 sides equal. Isosceles = exactly 2 sides equal. Scalene = no sides equal.

Proposition 1 constructs an equilateral triangle—the most symmetric kind.

Right triangle = has a 90° angle. Obtuse = has an angle > 90°. Acute = all angles < 90°.

A triangle can only have at most one right or obtuse angle.

Square = 4 equal sides + 4 right angles. Rectangle (oblong) = 4 right angles but sides not all equal. Rhombus = 4 equal sides but not right angles. Parallelogram (rhomboid) = opposite sides equal. Everything else = trapezoid.

Euclid's categories are slightly different from modern ones.

Parallel lines are in the same plane and never meet, no matter how far you extend them.

This definition sets up the famous Parallel Postulate (P5). The nature of parallel lines was debated for 2000 years and led to non-Euclidean geometry.