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ABCD acute altitude antecedent appears base bisect centre chord circle circumference circumscribed coincide common consequently construction contained converse corollary definition demonstration described diagonals diameter difference distance divided double draw elements enveloping equal equimultiples equivalent established Euclid exceed extremities figure follows former four geometry given greater half hence hypothesis included inscribed intersect isosceles join latter less line drawn longer magnitudes manner mean measure meet method middle multiple obvious opposite parallel perimeter perpendicular polygon portion PROBLEM produced Prop proportion proposed PROPOSITION proved quadrilateral radii radius reasoning rectangle regular remaining respectively equal rhomboid right angles Scholium shorter shown sides similar space square straight line subtended supposed surface taken tangent term THEOREM third triangle ABC twice vertex VIII whole
Side 159 - ... if a straight line, &c. QED PROPOSITION 29. — Theorem. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another ; and the exterior angle equal to the interior and opposite upon the same side ; and likewise the two interior angles upon the same side together equal to two right angles.
Side 80 - IF a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those produced, proportionally; and if the sides, or the sides produced, be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle...
Side 179 - FBC ; and because the two sides AB, BD are equal to the two FB, BC, each to each, and the angle DBA equal to the angle FBC ; therefore the base AD is equal to the base FC, and the triangle ABD to the triangle FBC.
Side 136 - To describe an isosceles triangle, having each of the angles at the base double of the third angle.
Side 179 - BK, it is demonstrated that the parallelogram CL is equal to the square HC. Therefore the whole square BDEC is equal to the two squares GB, HC ; and the square BDEC is described upon the straight line BC, and the squares GB, HC upon BA, AC.
Side 99 - And since a radius drawn to the point of contact is perpendicular to the tangent, it follows that the angle included by two tangents, drawn from the same point, is bisected by a line drawn from the centre of the circle to that point ; for this line forms the hypotenuse common to two equal right angled triangles. PROP. XXXVII. THEOR. If from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it ; if the rectangle...
Side 29 - In any triangle, the square of a side opposite an acute angle is equal to the sum of the squares of the other two sides diminished by twice the product of one of those sides and the projection of the other side upon it.
Side 179 - FC, and the triangle ABD to the triangle FBC. Now the parallelogram BL is double...