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ABCD adjacent angle ABC angle BAC angle BCD angle equal base BC BC is equal bisect centre chord circle ABC circumference coincide common Const CONSTRUCTION cqual describe diagonal diameter difference divided double draw drawn equal to CD exterior angle extremities fall four given point given straight line gnomon greater half impossible join length less Let ABC Let the straight manner meet opposite angles parallel parallelogram pass perpendicular possible Problem produced PROOF PROOF.—Because Proposition proved rectangle contained right angles segment semicircle shown side BC sides square on AC Take taken third touches the circle triangle ABC twice the rectangle unequal
Page 35 - IF a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are equal to two right angles.
Page 13 - THE angles at the base of an isosceles triangle are equal to one another : and, if the equal sides be produced, the angles upon the other side of the base shall be equal. Let ABC be an isosceles triangle, of which the side AB is equal to AC, and let the straight lines AB, AC be produced to D and E.
Page 7 - A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
Page 17 - If two triangles have two sides of the one equal to two sides of the...
Page 51 - If the square described upon one of the sides of a triangle, be equal to the squares described upon the other two sides of it ; the angle contained by these two sides is a right angle.
Page 9 - If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles...
Page 69 - To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, may be equal to the square of the other part.
Page 9 - Things which are double of the same, are equal to one another. 7. Things which are halves of the same, are equal to one another.
Page 32 - Wherefore, if a straight line, &c. QED PROPOSITION XXVIII. THEOREM. If a straight line falling upon two other straight lines, make the exterior angle equal to the interior and opposite upon the same side of the line ; or make the interior angles upon the same side together equal to two right angles ; the two straight lines shall be parallel to one another.
Page 67 - If a straight line be bisected, and produced to any point; the rectangle contained by the whole line thus produced, and the part of it produced, together with the square of half the line bisected, is equal to the square of the straight line, which is made up of the half and the part produced.