Triangles which are on the same base and in the same parallels equal one another. I say that the triangle abc equals the triangle dbc. 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. A semicircle is the figure contained by the diameter and the circumference cut off by it. This is the thirty fourth proposition in euclids first book of the elements. On a given straight line to construct an equilateral triangle. 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. In parallelograms, the opposite sides are equal, and the opposite angles are equal. If in a triangle two angles be equal to one another, the sides which subtend the equal angles will. 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.
New technologies for the study of euclids elements citeseerx. In isosceles triangles the angles at the base are equal to one another, and, if the equal straight. Euclid could have bundled the two propositions into one. Euclids elements of geometry, book 1, proposition 5 and book 4, proposition 5.
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