P l a n e G e o m e t r y An Adventure in Language and Logic based on Introduction. Geometry: The study of figures
Introduction to Logic. Hypothesis and conclusion
BOOK I First Principles. Definitions, Postulates, Axioms or CONSTRUCTIONS On a given straight line to construct an equilateral triangle. From a given point to draw a straight line equal to a given straight line. Given two unequal straight lines, to cut off from the longer line CONGRUENT TRIANGLES Proposition 4 (Side-Angle-Side) If two triangles have two sides equal to two sides respectively, and if the angles contained by those sides are also equal, then the remaining side will equal the remaining side, the triangles themselves will be equal areas, and the remaining angles will be equal, namely those that are opposite the equal sides. THE ISOSCELES TRIANGLE In an isosceles triangle the angles at the base are equal. PROOF BY CONTRADICTION If two angles of a triangle are equal, then the sides opposite them will be equal. CONGRUENT TRIANGLES 2 Proposition 8 (Side-Side-Side) If two triangles have two sides equal to two sides respectively, and if the bases are also equal, then the angles will be equal that are contained by the two equal sides. BISECTIONS To bisect a given angle. To bisect a given straight line. PERPENDICULARS To draw a straight line at right angles to a given straight line from a given point on it. To a given straight line that may be made as long as we please, and from a given point not on it, to draw a perpendicular line. RIGHT ANGLES, VERTICAL ANGLES When a straight line that stands on another straight line makes two angles, either it makes two right angles, or it makes angles that together are equal to two right angles. If two straight lines are on opposite sides of a given straight line, and, meeting at one point of that line they make the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another. When two straight lines intersect one another, the vertical angles are equal. THE SIDES AND ANGLES OF A TRIANGLE If one side of a triangle is extended, then the exterior angle is greater than either of the opposite interior angles. Any two angles of a triangle are together less than two right angles. A greater side of a triangle is opposite a greater angle. A greater angle of a triangle is opposite a greater side. Any two sides of a triangle are together greater than the third side To construct a triangle whose sides are equal to three given straight lines: thus any two of them taken together must be greater than the third. On a given straight line and at a given point on it, to construct an angle equal to a given angle. CONGRUENT TRIANGLES 3 Proposition 26 (Angle-Side-Angle) If two triangles have two angles equal to two angles respectively, and one side equal to one side, which may be either the sides between the equal angles or the sides opposite one of them, then the remaining sides will equal the remaining sides (those that are opposite the equal angles), and the remaining angle will equal the remaining angle. THE THEORY OF PARALLEL LINES If a straight line that meets two straight lines makes the alternate angles equal, then the two straight lines are parallel. If a straight line that meets two straight lines makes an exterior angle equal to the opposite interior angle on the same side, or if it makes the interior angles on the same side equal to two right angles, then the two straight lines are parallel. If two straight lines are parallel, then a straight line that meets them makes the alternate angles equal, it makes the exterior angle equal to the opposite interior angle on the same side, and it makes the interior angles on the same side equal to two right angles. Straight lines that are parallel to the same straight line are parallel THE THREE ANGLES OF A TRIANGLE Through a given point to draw a straight line parallel to a given straight line. If one side of a triangle is extended, then the exterior angle is equal to the two opposite interior angles; and the three interior angles of a triangle are equal to two right angles. PARALLELOGRAMS The straight lines which join the extremities on the same side of two equal and parallel straight lines, are themselves equal and parallel. In a parallelogram the opposite sides and angles are equal, and the diagonal bisects the area. EQUALITY OF NON-CONGRUENT FIGURES Parallelograms on the same base and in the same parallels are equal. Parallelograms on equal bases and in the same parallels are equal. Triangles on the same base and in the same parallels are equal. Triangles on equal bases and in the same parallels are equal. Equal triangles that are on the same base and on the same side of it, are in the same parallels. If a parallelogram and a triangle are on the same base and in the same parallels, the parallelogram is double the triangle. CONSTRUCTION OF A SQUARE On a given straight line to draw a square. THE PYTHAGOREAN THEOREM In a right triangle the square drawn on the side opposite the right angle If the square drawn on one side of a triangle is equal to the squares drawn on the other two sides, then the angle contained by those two sides is a right angle. |