Posts tagged 'book-i'
G.1 point
A point A is that which has no part.
G.2 line
A line [ A B ] is a projection of a relationship of two points A, B.
G.3 extremities of a segment
The extremities of a segment / A B / are points A, B.
G.4 segment
A segment / A B / is a portion of a line [ A B ] between two points A, B on the line.
G.5 surface
A surface is that which has length and breadth only.
G.6 extremities of a surface
The extremities of a surface are lines or arcs.
G.7 plane surface
A plane surface lies evenly with the lines on itself.
G.8 boundary
A boundary is the extremity of anything.
G.9 figure
A figure is that which is contained by any boundary or boundaries.
G.10 circle
A circle ( A B ) is a plane figure contained by one arc defined by a center A and a point B establishing its defining radius / A B /, such that all segments / A B /, / A C /, / A D / falling upon the arc from A are equal.
G.11 center of a circle
The point A within the circle ( A B ) from which all segments / A B /, / A C / to the arc are equal is called the center of the circle.
G.12 rectilineal figures
Rectilineal figures are contained by segments: trilateral by three, quadrilateral by four, and multilateral by more than four.
G.13 equilateral, isosceles, and scalene triangles
Of trilateral figures, an equilateral triangle < A B C > has three equal sides, an isosceles triangle < D E F > has two equal sides, and a scalene triangle < G H K > has three unequal sides.
G.14 construct a segment
Construct a segment / A B / between any two points A, B.
G.15 construct a circle
Construct a circle ( A B ) with center A and defining radius point B.
G.16 equals to the same are equal
Things equal to the same thing are equal to one another.
G.17 construct an equilateral triangle
Construct an equilateral triangle on a given segment.
G.18 extend a segment
Extend a segment / A B / continuously along a line [ A B ] to / B C /.
G.19 equals subtracted from equals are equal
If equals are subtracted from equals, the remainders are equal.
G.20 transfer a segment to a point
Construct at a given point a segment equal to a given segment.
G.21 cut off an equal segment
Cut off from the greater of two unequal segments a segment equal to the less.
G.22 plane angle
A plane angle is the inclination to one another of two lines or arcs in a plane that meet and do not lie on the same line.
G.23 coinciding things are equal
Things / A B /, / D E / that coincide with one another are equal to one another.
G.24 congruent triangles by SAS
Triangles with two sides and the included angle equal are congruent (SAS).
G.25 base angles of an isosceles triangle
In an isosceles triangle, the angles at the base are equal, and if the equal sides are extended, the angles under the base are equal.
G.26 the whole is greater than the part
The whole is greater than the part.
G.27 triangle with equal base angles is isosceles
If two angles of a triangle are equal, the sides subtending the equal angles are also equal.
G.28 uniqueness of triangle sides on a base
On the same segment and on the same side, there cannot be constructed two distinct triangles with both pairs of conterminous sides equal.
G.29 congruent triangles by SSS
Triangles with all three corresponding sides equal have their corresponding angles equal (SSS).
G.30 rectilineal angle
A rectilineal angle is contained by two segments or lines.
G.31 bisect an angle
Bisect a given angle.
G.32 bisect a segment
Bisect a given segment.
G.33 right angle and perpendicular
When a segment standing on a line makes the adjacent angles equal, each of the equal angles is a right angle, and the standing segment is perpendicular to the line.
G.34 construct a perpendicular at a point on a line
Construct a segment at right angles to a given line from a given point on it.
G.35 construct a perpendicular from a point to a line
Construct a perpendicular to a given line from a given point not on it.
G.36 equals added to equals are equal
If equals are added to equals, the wholes are equal.
G.37 adjacent angles on a line sum to two right angles
A segment standing on a line makes either two right angles or adjacent angles summing to two right angles.
G.38 right angles are equal
All right angles are equal.
G.39 supplementary adjacent angles form a line
If two segments at a point on a line on opposite sides make adjacent angles summing to two right angles, the two segments are collinear.
G.40 vertical angles are equal
Intersecting lines make their vertical angles equal.
G.41 exterior angle of a triangle is greater than interior
In any triangle, if one side is extended, the exterior angle is greater than either opposite interior angle.
G.42 two angles of a triangle are less than two right angles
In any triangle, the sum of any two angles is less than two right angles.
G.43 greater side subtends greater angle
In any triangle, the greater side subtends the greater angle.
G.44 greater angle is subtended by greater side
In any triangle, the greater angle is subtended by the greater side.
G.45 triangle inequality
In any triangle, the sum of any two sides is greater than the remaining side.
G.46 interior segments of a triangle
Segments constructed from the endpoints of one side of a triangle to an interior point have a lesser sum but contain a greater angle than the other two sides.
G.47 construct a triangle from three segments
Construct a triangle from three segments equal to three given segments satisfying the triangle inequality.
G.48 construct an angle equal to a given angle
Construct on a given line at a given point an angle equal to a given angle.
G.49 hinge theorem
If two triangles have two sides equal to two sides respectively, the one with the greater included angle has the greater base.
G.50 converse hinge theorem
If two triangles have two sides equal to two sides respectively, the one with the greater base has the greater included angle.
G.51 congruent triangles by ASA and AAS
Triangles with two angles and one corresponding side equal are congruent (ASA / AAS).
G.52 parallel lines
Parallel lines are lines in the same plane which, extended indefinitely in both directions, do not meet.
G.53 equal alternate interior angles imply parallel lines
If a transversal cutting two lines makes the alternate interior angles equal, the lines are parallel.
G.54 equal corresponding angles imply parallel lines
If a transversal cutting two lines makes the exterior angle equal to the opposite interior angle, or interior angles on the same side supplementary, the lines are parallel.
G.55 parallel postulate
Two lines intersected by a transversal with interior angles on one side summing to less than two right angles meet on that side.
G.56 transversal angles of parallel lines
A transversal cutting parallel lines makes alternate interior angles equal, exterior and opposite interior angles equal, and same-side interior angles supplementary.
G.57 lines parallel to the same line are parallel
Lines parallel to the same line are parallel to one another.
G.58 construct a line parallel to a given line
Construct through a given point a line parallel to a given line.
G.59 exterior angle and angle sum of a triangle
In any triangle, an exterior angle equals the sum of the two opposite interior angles, and the three interior angles sum to two right angles.
G.60 segments joining equal and parallel segments
Segments joining the endpoints of equal and parallel segments in the same directions are themselves equal and parallel.
G.61 diameter of a circle
A diameter of a circle is any segment drawn through the center and terminated in both directions by the arc of the circle; it bisects the circle.
G.62 opposite sides, angles, and diagonal of a parallelogram
In a parallelogram, opposite sides and opposite angles are equal, and the diagonal bisects the area.
G.63 square, oblong, rhombus, and rhomboid
Of quadrilateral figures, a square is equilateral and right-angled; an oblong is right-angled but not equilateral; a rhombus is equilateral but not right-angled; a rhomboid has opposite sides and angles equal but is neither equilateral nor right-angled.
G.64 parallelograms on the same base in the same parallels
Parallelograms on the same base and in the same parallels are equal in area.
G.65 parallelograms on equal bases in the same parallels
Parallelograms on equal bases and in the same parallels are equal in area.
G.66 triangles on the same base in the same parallels
Triangles on the same base and in the same parallels are equal in area.
G.67 triangles on equal bases in the same parallels
Triangles on equal bases and in the same parallels are equal in area.
G.68 equal triangles on the same base are in the same parallels
Equal triangles on the same base and on the same side lie in the same parallels.
G.69 equal triangles on equal bases are in the same parallels
Equal triangles on equal bases and on the same side lie in the same parallels.
G.70 parallelogram is double a triangle on the same base
If a parallelogram shares the same base with a triangle and lies in the same parallels, the parallelogram is double the triangle.
G.71 construct a parallelogram equal to a triangle
Construct, in a given angle, a parallelogram equal in area to a given triangle.
G.72 complements of parallelograms about the diagonal are equal
In any parallelogram, the complements of the parallelograms about the diagonal are equal in area.
G.73 apply a parallelogram equal to a triangle on a segment
Apply to a given segment, in a given angle, a parallelogram equal in area to a given triangle.
G.74 construct a parallelogram equal to a polygon
Construct, in a given angle, a parallelogram equal in area to a given polygon.
G.75 construct a square on a segment
Construct a square on a given segment.
G.76 obtuse angle
An obtuse angle is greater than a right angle.
G.77 acute angle
An acute angle is less than a right angle.
G.78 right, obtuse, and acute-angled triangles
A right-angled triangle has a right angle, an obtuse-angled triangle has an obtuse angle, and an acute-angled triangle has three acute angles.
G.79 Pythagorean theorem
In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the two legs (Pythagorean Theorem).
G.80 converse of the Pythagorean theorem
If the square on one side of a triangle equals the sum of the squares on the other two sides, the angle contained by those two sides is right.
G.96 semicircle
A semicircle is the figure contained by a diameter and the arc cut off by it.
Defined-Term Dependency Wiring, Short Verb Titles & Two-Point Circle Notation
Released geometor.elements v0.4.0, integrating defined-term dependency wiring across all 13 books (G.1 through G.601), concise lowercase verb-phrase titles, foundational SVG constructions, the ≃ coincidence operator, and the two-point circle convention ( A B ).
G-Index Book I (G.1–G.89) Three-Track Codex & Symbolic Indications
Released geometor.elements v0.3.3, completing the 3-track G-Index Book Codex (G.1 through G.89) for Book I of Euclid’s Elements.