For any parallelogram, the complements of the parallelograms about the diagonal are equal to one another.
Let ABCD be a parallelogram, and AC its diagonal. And let EH and FG be the parallelograms about AC,and BK and KD the so-called complements (about AC). I say that the complement BK is equal to the complement KD.
For since ABCD is a parallelogram, and AC its diagonal, triangle ABC is equal to triangle ACD [Prop. 1.34]. Again, since EH is a parallelogram, and AK is its diagonal, triangle AEK is equal to triangle AHK [Prop. 1.34]. So, for the same (reasons), triangle KFC is also equal to (triangle) KGC. Therefore, since triangle AEK is equal to triangle AHK, and KFC to KGC, triangle AEK plus KGC is equal to triangle AHK plus KFC. And the whole triangle ABC is also equal to the whole (triangle) ADC. Thus, the remaining complement BK is equal to the remaining complement KD.
Thus, for any parallelogramic figure, the complements of the parallelograms about the diagonal are equal to one another. (Which is) the very thing it was required to show.
"As for ancient geometrical analysis and modern algebra, even apart from the fact that they deal only in highly abstract matters that seem to have no practical application, the former is so closely tied to the consideration of figures that it is unable to exercise the intellect without greatly tiring the imagination, while in the latter case one is so much a slave to certain rules and symbols that it has been turned into a confused and obscure art that bewilders the mind instead of being a form of knowledge that cultivates it."
— R. Descartes, A Discourse on the Method, 1637
"... geometry is nothing other than that part of universal mechanics which reduces the art of measuring to exact propositions and demonstrations. But geometry is commonly used in reference to magnitude, and mechanics in reference to motion. In this sense rational mechanics will be the science, expressed in exact propositions and demonstrations, of the motions that result from any forces whatever and of the forces that are required for any motions whatever."
— I. Newton, Preface to Principia, 1687
Start with Definitions, Postulates, Common Notions or First proposition.
Interesting proofs:
Complements in parallelograms,
Pythagorean theorem
Start with Definitions or First proposition.
Interesting construction:
Golden section
Start with Definitions or First proposition.
Interesting propositions:
Angles at center and circumference,
Relations of lines from outside a circle
Start with Definitions or First proposition.
Interesting construction: Regular Pentagon
Start with Definitions or First proposition.
Interesting proof:
Commutativity of multiplication
Start with Definitions or First proposition.
Interesting propositions:
Ratios of similar areas,
Solution of a quadratic equation
Start with Definitions or First proposition.
Interesting propositions: Greatest common divisor algorithm
Start with First proposition.
Interesting proof: Prime factors in series
Start with First proposition.
Interesting proof: Infinity of primes
Start with Definitions I, Definitions II or Definitions III.
Interesting proof: Method of exhaustion
Start with Definitions or First proposition.
Interesting construction: Solid angle
Start with First proposition.
Interesting proof: Pyramid as the third of a prism
Start with First proposition.
Interesting propositions: Icosahedron, Dodecahedron, Finitude of regular polyhedra
Based on this translation.
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Start with Definitions, Postulates, Common Notions or First proposition.
Interesting proofs:
Complements in parallelograms,
Pythagorean theorem
Start with Definitions or First proposition.
Interesting construction:
Golden section
Start with Definitions or First proposition.
Interesting propositions:
Angles at center and circumference,
Relations of lines from outside a circle
Start with Definitions or First proposition.
Interesting construction: Regular Pentagon
Start with Definitions or First proposition.
Interesting proof:
Commutativity of multiplication
Start with Definitions or First proposition.
Interesting propositions:
Ratios of similar areas,
Solution of a quadratic equation
Start with Definitions or First proposition.
Interesting propositions: Greatest common divisor algorithm
Start with First proposition.
Interesting proof: Prime factors in series
Start with First proposition.
Interesting proof: Infinity of primes
Start with Definitions I, Definitions II or Definitions III.
Interesting proof: Method of exhaustion
Start with Definitions or First proposition.
Start with First proposition.
Interesting proof: Pyramid as the third of a prism
Start with First proposition.
Interesting propositions: Icosahedron, Dodecahedron, Finitude of regular polyhedra
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