Proposition XI. Problem.
|(72)||From a given point (C) in a given right line (A B) to draw a perpendicular to the given line.|
Because the sides D F and D C are equal to the sides E F and E C (const.), and C F is common to the triangles D F C and E F C, therefore (VIII) the angles opposite to the equal sides D F and E F are equal, and therefore F C is perpendicular to the given right line A B at the point C.
Cor.—By help of this problem it may be demonstrated, that two straight lines cannot have a common segment.
It it be possible, let the two straight lines A B C, A B D have the segment A B common to both of them. From the point B draw B E at right angles to A B; and because A B C is a straight line, the angle C B E is equal to the angle E B A; in the same manner, because A B D is a straight line, the angle D B E is equal to the angle E B A; wherefore the angle D B E is equal to the angle C B E, the less to the greater, which is impossible; therefore the two straight lines cannot have a common segment.
If the given point be at the extremity of the given right line, it must be produced, in order to draw the perpendicular by this construction.
In a succeeding article, the student will find a method of drawing a perpendicular through the extremity of a line without producing it.
The corollary to this proposition is useless, and is omitted in some editions.
It is equivalent to proving that a right line cannot be produced through its extremity in more than one direction, or that it has but one production.
Book I: Euclid, Book I (ed. Dionysius Lardner, 11th Edition, 1855)
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