Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/40

 PROPOSITION 11. PROBLEM. To draw a straight line at right angles to a given straight line, from a given point in the same. Let AB be the given straight line, and C the given point in it: it is required to draw from the point C a straight line at right angles to AB. Take any point D in AC and make CE equal to CD On DE describe the equilateral triangle DFE, and join CF. The straight line CF drawn from the given point C shall be at right angles to the given straight line AB. Because DC is equal to CE, and CF is common to the two triangles DCF, ECF; the two sides DC, CF are equal to the two sides EC, CF, each to each; and the base DF is equal to the base EF; therefore the angle DCF is equal to the angle ECF; and they are adjacent angles. But when a straight line, standing on another straight line, makes the adjacent angles equal to one another, each of the angles is called a right angle; therefore each of the angles DCF, ECF is a right angle. Wherefore from the given point C in the given straight line AB, CF has been drawn at right angles to AB. Corollary. By the help of this problem it may be shewn that two straight lines cannot have a common segment. If it be possible, let the two straight lines ABC, ABD have the segment AB common to both of them. From the point B draw BE at right angles to AB. Then, because ABC is a straight line, the angle CBE is equal to the angle EBA.