A quick trip through the Elements

To get an idea of what's in the Elements, here are a few highlights in the order that they appear:

I.Def.23, definition of parallel lines, one of many definitions in Book I
I.Post.5, the parallel postulate
Common notions, the axioms for magnitudes
I.1, the first proposition which shows how to construct an equilateral triangle
The congruence theorems for triangles: I.4, side-angle-side, I.8, side-side-side, and I.26, angle-side-angle
Propositions on isosceles triangles: I.5, equal angles imply equal sides, and the converse, I.6, equal sides imply equal angles
I.9 and I.10, constructions to bisect angles and line segments
I.11 and I.12, constructions to draw perpendicular lines
I.16, an exterior angle of a triangle is greater than either of the opposite interior angles (compare I.32 below)
I.29, about angles made when a line crosses two parallel lines
I.20, the triangle inequality (the sum of two sides is greater than the third)
I.22, to construct a triangle with given sides
I.32, an exterior angle of a triangle is the sum of the two opposite interior angles; the sum of the three interior angles equals two right angles.
On application of areas: I.42 to find a parallelogram equal in area to any given triangle, and I.45 to find a parallelogram equal in area to any given polygon
I.47, the Pythagorean theorem and its converse I.48

II.4, a geometric version of the algebraic identity (x + y)² = x² + 2xy + y²
II.5, a sample proposition showing how to factor the difference of two squares
II.6, a geometric version to solve the quadratic equation (b – x)x = c
II.11, construction to cut a line in the golden ratio
II.12 and II.13, a pre-trigonometry version of the law of cosines
II.14, a final proposition on application of areas—to find a square equal in area to any given polygon

III.1, how to find the center of a circle
III.17, how to draw a line tangent to a circle
Propositions on angles in circles: III.20, III.21, and III.22
III.31, Thales' theorem that an angle inscribed in a semicircle is right, and similar statements giving acute and obtuse angles
III.35, when two chords are drawn through a point inside a circle, then the product of the two segments of one chord equals the product of the two segments of the other chord
III.36, if from a point outside a circle both a tangent and a secant are drawn, then the square of the tangent is the product of the whole secant and the external segment of the secant, and the converse in III.37

Inscribing and circumscribing circles and arbitrary triangles IV.2, IV.3, IV.4, and IV.5,
IV.10, how to construct a particular triangle needed for regular pentagons
IV.11, how to construct a regular pentagon
IV.15, how to construct a regular hexagon
IV.16, how to construct a regular 15-gon

VI.Def.1, definition of similar figures
VI.1, areas of triangles (also parallelograms) of the same height are proportional to their bases
VI.2, a line parallel to the base of a triangle cuts the sides proportionally
Propositions on similar triangles: VI.4, VI.5,
VI.6, side-angle-side similarity theorem
VI.9, to cut a line into a given number of equal segments
VI.10, to cut a line into a specified ratio
Constructions of fourth proportionals VI.12, and mean proportionals VI.13,
VI.16, if four lines are proportional, w : x = y:z, then the rectangle contained by the extremes, w by z, has the same area as the rectangle contained by the means, x by y
VI.19, on areas of similar triangles
VI.25, on application of areas
VI.31, a generalization of the Pythagorean theorem to figures other than squares

VII.Def.11, definition of prime number
VII.12, the Euclidean algorithm for finding greatest common divisors
Several basic properties of numbers, such as VII.16, commutativity of multiplication of numbers, mn = nm.
VII.29, if a prime number doesn't divide a number, then it is relatively prime to it
VII.30, if a prime number divides a product of two numbers, then it divides one of them
VII.34, on constructing least common multiples

VIII.2 and VIII.4, on finding continued proportions of numbers
Many propositions on squares and cubes, such as VIII.22, if three numbers are in continued proportion, and the first is square, then the third is also square

IX.14, a partial version of the fundamental theorem of arithmetic that says no prime number can divide a product of other prime numbers
IX.20, there are infinitely many prime numbers
Several propositions on even and odd numbers, such as IX.23 which says that if you add an odd number of odd numbers together, then the sum is odd
IX.35, how to get the sum of a geometric progression
IX.36, on perfect numbers

XII.2, areas of circle are proportional to the squares on their diameters
XII.6 and XII.7, a triangular prism can be divided into three pyramids of equal volume, hence, the volume of a pyramid is one third of that of the prism with the same base and same height
XII.10, the volume of a cone is one third of that of the cylindar with the same base and same height
XII.11, volumes of cones and cylinders are proportional to their heights
XII.18, on volumes of spheres

XIII.9, on hexagons and decagons inscribed in a circle, and the golden ratio
XIII.10, on hexagons and decagons inscribed in a circle, and the golden ratio
XIII.11, when a pentagon, hexagon, and decagon are inscribed in a circle, the square on the side of the pentagon equals the sum of the squares on the sides of the hexagon and the decagon
Constructions of regular polyhedra XIII.13, XIII.15, XIII.14, XIII.16, and XIII.17
These five are shown to be the only regular solids in proposition XIII.18.