# Book I Dependencies

At the bottom of this page will see a dependency chart for the propositions of Book I. Proposition I starts at the bottom of the image in green. The chart is available for anyone to use from: pdf file.

The colored groups refer to the classification of the propositions in the Book: Euclid's Elements Book I, a new rendering.

The three top propositions are:

Proposition 48, which is the converse of the famous Pythagorean theorem (Prop 47) is the topmost theorem in Book I - in blue to the right.

If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right.

Proposition 47, which is just beneath Prop 48 is the famous Pythagorean Theorem that every middle school and high school student knows.

In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.

Proposition 45, is the the top most and perhaps the most important but much less well-known theorem. The theorem lets one define the area of any rectangular shape by converting it into a parallelogram.

To construct a parallelogram equal to a given rectilinear figure in a given rectilinear angle.

The colored groups are as follows:

Group 1: Equilateral triangles, basic manipulation theorems, and SAS theorem.

Group 2: Isosceles Triangles

Group 3: SSS Theorem

Group 4: Bisection and Perpendicular Theorem

Group 5: Opposite Angles

Group 6: Properties of Triangles

Group 7: Parallel Lines

Group 8: Angles of Triangles

Group 9: Parallelogram and Triangle Theorems Related to Areas

Group 10: Area Arithmetic Propositions

Group 11: Pythagoras’ Theorem