fundamentals

defining and constructing the relationships of points, lines, circles

• definitions
  • plane
  • point
  • line
  • segment
  • circle
    • diameter
    • arc
    • semicircle
  • angle
    • acute angle
    • obtuse angle
    • right angle
  • boundary
  • figure
    • rectilineal figures
    • trilateral figures
    • quadrilateral figures
  • right triangle
  • parallel lines
• common notions
  • Things which are equal to the same thing are also equal to one another
  • If equals be added to equals, the wholes are equal
  • If equals be subtracted from equals, the remainders are equal
  • Things which coincide with one another are equal to one another
  • The whole is greater than the part
• postulates
  • set point
  • construct line
  • construct circle
  • set segment
  • set polygon
  • right angles are equal
  • non parallel lines
• propositions
  • construct an equilateral triangle on a segment
    • dependencies
    • steps
  • construct equal segments by extension
    • dependencies
    • footnotes
  • construct equal segments by section
    • dependencies
  • triangles with equal sides have equal angles
    • dependencies
  • The base angles of an isosceles triangle are equal
    • dependencies
    • footnotes
  • when two angles of a triangle are equal, so are the subtended sides
  • the sides of a triangle are uniquely related to the position of the vertexes
    • references
    • footnotes
  • triangles with equal sides have equal angles
    • dependencies
    • footnotes
  • bisect angle
    • references
  • bisect segment
  • construct perpendicular from point on line
    • references
    • footnotes
  • construct perpendicular from point not on line
    • references
    • footnotes
  • angles from intersecting lines equal two right angles (on one side)
    • footnotes
  • adjacent angles equal to two right angles make a line
  • opposing angles of intersecting lines are equal
    • porism
    • references
    • footnotes
  • the sum of any two angles of a triangle is less than two right angles
    • references
    • footnotes
  • the greater side of a triangle subtends the greater angle
  • the greater angle of a triangle is subtended by the greater side
    • references
  • any two sides of a triangle are greater than the remaining side
    • references
    • footnotes
  • Proposition I.21
    • references
    • footnotes
  • construct a triangle from three segments
    • references
    • footnotes
  • construct a given angle on a given segment
    • references
  • similar triangles
    • references
    • footnotes
  • similar triangles
    • references
  • similar triangles
    • references
    • footnotes
  • the angles produced from a line intersecting parallel lines are equal
  • similar angles from line intersecting parallels
    • references
  • parallel angles
    • references
    • footnotes
  • parallels are parallel to each other
    • references
    • footnotes
  • construct a parallel to a line through a point
    • references
  • similar triangles
    • references
  • similar angles on parallels
    • references
    • footnotes
  • parallelogram angles
    • references
    • footnotes
  • parallelogram equality
  • parallelogram equality
  • triangles in parallels equality
  • triangle and parallels equality
  • triangles and parallels
  • triangles and parallels
  • parallelograms and triangles
  • construct a parallelogram equal to a given triangle on a given angle
  • equality of parallelogram complements
  • construct a parallelogram on a segment equal to a given triangle
  • construct a parallelogram equal to a given polygon on a given angle
  • construct a square
  • the squares on the sides of a right triangle are equal to the square on the hypotenuse
  • a triangle is right if the sum of squares of two sides equals the square on the remaining