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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Lemma prop_12 : forall A B C, ~ Col A B C -> exists H, Col A B H /\ Perp A B C H.