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Given. The straight line AB and the point C in A B.
Required. To draw from the point Ca straight line at right angles to AB.
(a) In A C take any point D.
(b) From CB cut off a part CE equal to CD. (Euc. I. 3.)
(c) On DE describe the equilateral triangle DFE. (Euc. I. 1.)
(d) Join FC.
The straight line FC drawn from the given
point C, shall be at right angles to the given straight line AB.
If FC be at right angles to A B, we must prove that angle F C D is equal to angle FCE. Proof (with Syllogisms in full).
DC equals CE (by Construction b); add to each CF.
(e) Then by Axiom 2a, DC, CF are equal to EC, CF, each to each.
If two triangles have two sides of the one equal to two sides of the other, and the base of the one equal to the base of the other; those sides shall contain equal angles (Euc. I. 8).
The two triangles FCD, FCE have the two sides D C, CF equal to the two sides E C, CF (e), each to each, and they have the base D equal to the base EF (by construction (c), FDC being an equilateral triangle).
(f).. the angle DC F is equal to the angle ECF.
When a straight line, standing on another straight line, makes the adjacent angles equal, each of the angles is called a right angle (Euc. Def. 10).
The straight line FC standing on the straight line AB makes the angle D CF equal to the adjacent angle ECF. (f).
.. Each of the angles D C F, E C F is called a right angle.
Result.-Wherefore from the given point C, a line CF has been drawn at right angles to A B.
EXERCISES.-I. Write out this proof in contracted syllogisms.
II. At the point in the given straight line NO draw a straight line at right angles to NO (produce NO towards N).
III. At the point NV in the straight line given in (II), draw a straight line double the length of NO at right angles to NO. (Produce NO both ways; make the produced parts each equal to NO; draw a line at right angles from N; describe a circle with N as centre and radius twice NO; produce line at right angles to NO till it meets the circle.)
PROBLEM (Euclid I. 12).
Repeat. The definition of a circle, and of a right angle and the enunciation of Euc. I. 8 (page 46). General Enunciation.
To draw a straight line perpendicular to a given straight line from a given point without it.
Given. The straight line
Required. To draw from C a A line perpendicular to A B. Construction.
Take any point D on the side of A B remote from C.
At the centre C and distance CD describe the circle FDG, cutting A Bin FG. (The whole circle is not shown in the figure, only the part needed.)
(a) Bisect FG (Euc. I. 10 shows how), and call the point of bisection H.
The straight line CH drawn from the given
point Cis perpendicular to the given straight line A B.
If CH is perpendicular to A B, the angle CHF will be equal to the angle CHG.
Then we must prove that angle CHF is equal to angle CH G.
Proof (with syllogisms in full).
Additional construction required for the purposes or proof. Join CF, CG.
All lines drawn from the centre of a circle to the circumference are equal. (Definition of a circle.)
CF and C G are drawn from the centre C to the circumference FDG.
(b)... C F is equal to C G.
FH is equal to HG (by Construction a). (c).. FH, HC are equal to G H, H C, each
If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the base of one triangle, equal to the base of the other, then those sides shall contain equal angles. (Euc. I. 8.)
The two triangles CFH, CGH have the two sides FH, HC equal to the two sides GH, FC (c), and the base CF equal to the base CHF (b).
(d) .. The angle CHF is equal to the angle CH G.
When a straight line standing on another straight line makes the adjacent angles equal to one another, the straight line which stands on the other is called a perpendicular to it (definition).
CH, standing on AB, makes the angle CHF equal to the adjacent angle CHG (d).
... CH is perpendicular to AB.
Result. Wherefore a perpendicular CH has been drawn to the given line AB from the given point C.
Q. E. F.
EXERCISES.-I. Write out the proof of this proposition, omitting the major premiss of each syllogism, and giving the definition or proposition of Euclid referred to as the authority for the minor premiss. [Thus the first syllogism will read: Because CF and CG are drawn from the centre C to the circumference FDG. .. CF is equal to CG. (Euc. Def. of circle.)].