RELFE BROTHERS' EUCLID SHEETS. PROPOSITIONS 1-26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS. PROPOSITION XXI. If from the ends of a side of a triangle, there be drawn two straight lines to a point within the triangle ; these shall be less than the other two sides of the triangle, but shall contain a greater angle. Let be a triangle, and from the points , the ends of the side let the two straight lines be drawn to a point within the triangle. Then and shall be less than and the other two sides of the triangle, but shall contain an angle greater than the angle Produce to meet the side in Because two sides of a triangle are greater than the third side, therefore the two sides of the triangle are greater than ; to each of these unequals add ; therefore the sides are greater than Again, because the two sides of the triangle ; add to each of these unequals; therefore the sides are greater than But it has been shown that are greater than ; much more then greater than are Again, because the exterior angle of a triangle is greater than the interior and opposite angle; therefore the exterior angle of the triangle is greater than the interior and opposite angle ; for the same reason the exterior angle of the triangle is greater than the interior and opposite angle ; and it has been demonstrated, that the angle is greater than the angle ; much more therefore is the angle greater than the angle Therefore, if from the ends of the side, &c. RELFE BROTHERS EUCLID SHEETS. ' PROPOSITIONS 1-26, BOOK I, ARE NOW PUBLISHED IN A SIMILAR FORM TO THIS. PROPOSITION XX. Any two sides of a triangle are greater together than the third side. Let be a triangle. Then any two sides of it together shall be greater than the third side, viz., the sides and that the greater angle is subtended by the greater side ; therefore the side is greater than the side ; but is equal to and therefore the sides are greater than ; in the same manner it may be demonstrated, that the sides are greater than ; also that are greater than Therefore any two sides, &c. |