To prove that, draw CF and CG. |
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Now, since FH is equal to HG, |
(Construction) |
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and HC is common to the two triangles FHC, GHC, |
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the two sides FH, HC are equal to the two sides GH, HC respectively; |
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and the base CF is equal to the base CG; |
(Definition 15) |
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therefore angle CHF is equal to angle CHG. |
(S.S.S.) |
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And they are adjacent angles. |
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And when a straight line that stands on another straight line makes the adjacent angles equal, each of those angles is a right angle, and the straight line that stands on the other is called a perpendicular to it. |
(Definition 3) |
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Therefore we have drawn CH perpendicular to the given straight line AB, from C the given point not on it. Q.E.F. |
A word about what we mean by a given line or a given point, because those expressions occur in many propositions. For something to be given in this science, we must in some way be able to recognize it or know it. "That one." A line will be given if we can reproduce it.
For a point to be given, we must be able to reproduce its position. "There." This is analogous to the idea of a given number, which is a number we can recognize by its name; and which, if it is not rational, we can approximate by a rational number as closely as we please.
The requirement of given points, lines, and numbers insures that we do not lose touch with what we actually know.
Please "turn" the page and do some Problems.
or
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