Nneuclid's elements book 1 proposition 34 constitution

On a given straight line to construct an equilateral triangle. I say that the triangle abc equals the triangle dbc. Euclids elements of geometry, book 1, proposition 5 and book 4, proposition 5. It is required to draw a straight line through the point a parallel to the straight line bc. Let a be the given point, and bc the given straight line. In an introductory book like book i this separation makes it easier to follow the logic, but in later books special cases are often bundled into the general proposition.

Let acdb be a parallelogrammic area, and bc its diameter. 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. If in a triangle two angles be equal to one another, the sides which subtend the equal angles will. If with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another. Let abc and dbc be triangles on the same base bc and in the same parallels ad and bc. Propositions 1 and 2 in book 7 of elements are exactly the famous eu. To draw a straight line through a given point parallel to a given straight line.

A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. In isosceles triangles the angles at the base are equal to one another, and, if the equal straight. A semicircle is the figure contained by the diameter and the circumference cut off by it. In parallelogrammic areas the opposite sides and angles equal one another, and the diameter bisects the areas. In parallelograms, the opposite sides are equal, and the opposite angles are equal. New technologies for the study of euclids elements citeseerx. This proof shows that within a parallelogram, opposite angles and. Triangles which are on the same base and in the same parallels equal one another. Euclid could have bundled the two propositions into one. I say that the opposite sides and angles of the parallelogram acdb equal one another, and the diameter bc bisects it. This is the thirty fourth proposition in euclids first book of the elements.