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The triangles CFH and CDI, being similar. 3. CF (or LH): FH :: CD : DI.
4. CD:DI:: CF (or LH): FH.
The triangles CDI and CBK are similar : for the angle CIB=KCB, being alternate ones (by part. 2. theo. 3.) the lines CB and DI being parallel : the angle CDI= CBK being both right, and consequently the angle DCI=CKB, wherefore,
5. DI : CD :: CB : BK.
And again, making use of the similar triangles CLH and CBK.
6. CL: CB : : CH : CK.
7. HL:CH:: BK: CK.
make a triangle of three given right lines
BO, LB, LO, of which any two must be greater than the third. fig. 7.
Lay BL from B to L; from B with the line BO, describe an arc, and from L with LO describe another arc; from O, the intersection point of those arcs, draw BO and OL, and BOL is the triangle required.
At a point B in a given right line BC, to make an angle equal to a given angle A, fig. 8.
Draw any right line ED to form a triangle, as EAD, take BF = AD, and upon BF make the triangle BFG, whose side BG=AE, and GF=ED (by the last) then also the angle B=A; if we suppose one triangle be laid on the other, the sides will mutually agree with each other, and therefore be equal; for if we consider these two triangles are made of the same given three lines, they are manifestly one and the same triangle.
Upon the centres A and B, at any distance, let two arcs, DE, FG, be described ; make the arc FG=DE, and through B and G draw the line BG and it is done.
For since the chords ED, GF are equal, the angles A and B are also equal, as before (by
To bisect or divide into two equal parts, any given right lined angle, BAC. fig. 9.
In the lines AB and AC, from the point A set off equal distances AE=AD, then, with any distance more than the half of DE, describe two arcs to cut each other in some point F; and the rightline AF, joining the points A and F, will bisect in the given angle BAC.
For if DE and FE be drawn, the triangles ADF, AEF, are equilateral to each other, viz. AD=AE, DF=FE, and AF common, wherefore DAF= EAF, as before.
To bisect a right line. AB. fig. 10.
With any distance, more than half the line, from A and B, describe two circles CFD, CGD, cutting each other in the points G and D; draw GD intersecting AB in E, then AE=EB.
For, if AC, AD, BC, BD, be drawn, the triangles ACD, BCD, will be mutually equilateral, and consequently the angle ACE=BČE: therefore the triangle AČE, BCE, having AC=BC, CE common, and the angle ACE=BCE: (by theo. 6.) the base AE-the base BE.
Cor. Hence it is manifest, that CD not only bisects AB, but is perpendicular to it. (by def. 11.)
On a given point A, in a right line, EF, to erect a perpendicular. fig. 11.
From the point A lay off on each side, the equal distances, AC, AD; and from C and D, as centres, with any interval greater than AC or AD, describe two arcs intersecting each other in B; from A to B draw the line AB, and it will be the perpendicular required.
For, let CB, and BD be drawn; then the triangles CAB, DAB, will be mutually equilateral and equiangular, so CAB=DAB, a right angle, (by
To raise a perpendicular on the end B of a right line AB. Fig. 12.
From any point D not in the line AB, with the distance from D to B, let a circle be described cutting AB in E; draw from E through D the right line EDC, cutting the periphery in C, and join CB; and that is the perpendicular required.
EBC being a semicircle, the angle EBC will be a right angle (by cor. 5. theo. 7.)
From a given point A, to let fall a perpendicular upon a given right line BC. fig. 13
From any point D, in the given line, take the distance to the given point A, and with it describe a circle AGE, make GE=AG; join the points A and E, by the line AFE, and AF will be the perpendicular required.
Let DA, DE, be drawn; the angles ADF=FDE, DA=DE, being radii of the same circle, and DF common; therefore (by theo. 6.) the angle DFA =DFE, and FA a perpendicular. (By def. 11.)
Through a given point A, to draw a right line AB, parallel to a given right line CD, fig. 14.
From the point A, to any point, F, in the line CD, draw the line AF; with the interval FA, and one foot in F, describe the arc AE, and with the like interval and one foot in A, describe the arc BF, making BF=AE; through A and B draw the line AB, and it will be parallel to CD.
By prob. 2. The angle BAF=AFE, and by theo. 11. BA and CD are parallel.