Section Book_1_part_2.
(** For the next 27 propositions, we do not need the 5th axiom of Euclid, nor any continuity axioms, except for Proposition 22, which needs Circle/Circle intersection axiom. *)
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
(** * Proposition 2 To place at a given point (as an extremity) a straight line equal to a given straight line. *)
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(** * Proposition 3 Given two unequal straight lines, to cut off from the greater a straight line equal to the less. *)
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(** * Proposition 4 If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend. *)
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(** * Proposition 5 In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another. *)
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(** * Proposition 6 If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another. *)
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(** * Proposition 7 Given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there cannot be constructed on the same straight line (from its extremities), and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same extremity with it. *)
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(** * Proposition 8 If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines. *)
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(** * Proposition 9 To bisect a given rectilineal angle. *)
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(** * Proposition 10 To bisect a given finite straight line. *)
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(** * Proposition 11 To draw a straight line at right angles to a given straight line from a given point on it. *)
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(** * Proposition 12 To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line. *)
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(** * Proposition 13 If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles. *)
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(** * Proposition 14 If with any straight line, and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be in a straight line with one another. *)
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(** * Proposition 15 If two straight lines cut one another, they make the vertical angles equal to one another. *)
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(** * Proposition 16 In any triangle, if one of the sides is produced, the exterior angle is greater than either of the interior and opposite angles. *)
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(** * Proposition 17 In any triangle two angles taken together in any manner are less than two right angles. *)
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(** Here, the fact that the two angles are less than two right angles is described with the SAMS predicate, which means that they have a "sum at most straight", and the fact that their sum is not a straight line. *)
(** * Proposition 18 In any triangle the greater side subtends the greater angle. *)
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(** * Proposition 19 In any triangle the greater angle is subtended by the greater side. *)
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(** * Proposition 20 In any triangle two sides taken together in any manner are greater than the remaining one. *)
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(** * Proposition 21 If on one of the sides of a triangle, from its extremities, there be constructed two straight lines meeting within the triangle, the straight lines so constructed will be less than the remaining two sides of the triangle, but will contain a greater angle. *)
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(** * Proposition 22 Out of three straight lines, which are equal to three given straight lines, to construct a triangle: thus it is necessary that two of the straight lines taken together in any manner should be greater than the remaining one (cf. [I. 20]). *)
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(** This needs Circle/Circle intersection axiom *)
(** * Proposition 23 On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle. *)
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(** * Proposition 24 If two triangles have the two sides equal to two sides respectively, but have the one of the angles contained by the equal straight lines greater than the other, they will also have the base greater than the base. *)
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(** * Proposition 25 If two triangles have the two sides equal to two sides respectively, but have the base greater than the base, they will also have the one of the angles contained by the equal straight lines greater than the other. *)
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(** * Proposition 26 If two triangles have the two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides equal to the remaining sides and the remaining angle to the remaining angle. *)
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(** * Proposition 27 If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another. *)
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Lemma prop_27 : forall A D E F, TS E F A D -> CongA A E F D F E -> Par E A F D.