OP perpendicular to GK. (Eucl. ii. 6.) the rectangle NG, GK together with the square of PN is equal to the square of PG; to each of these add the square of PO; and the rectangle NG, GK together with the squares of OP, PN (i. e. the square of OC) is equal to the squares of OP, PG, i. e. to the square of OG, or to the squares of OH, HG. But the square of OH is equal to the rectangle CH, HA together with the square of OC; whence the rectangle NG, GK is equal to the rectangle CH, HA together with the square of HG. In the same manner it may be shewn that the rectangle LG, GM is equal to the rectangle DH, HB together with the square of HG. But since the rectangle NG, GK, is equal to the rectangle LG, GM, the rectangle CH, HA is equal to the rectangle DH, HB.

COR. If IH be a mean proportional between CH and HA; IG=GE.

(44.) Through the point of intersection of two given circles, to draw a line in such a manner that the sum of the respective rectangles contained by the parts thereof which are intercepted between the said point and their circumferences, and given lines A and B, may be equal to a given square.

Let the two circles CID, CEK cut each other in the point C; from C draw the diameters CD, CE. In CD take the point Fsuch, that CD: CF: A B. Join EF; and on

:

D

H

K

it as a diameter describe a semicircle, in which place EG a third proportional to A and the side of the given square. Draw ICK parallel to EG; it will be the line required.

Join FG, and produce it to H. The angle DIC is

equal to FGE, i. e. to FHC, .. FH is parallel to DI;

and CI CH :: CD: CF :: A : B,

... the rectangle A, CH, is equal to the rectangle B, CI. Now since EG is a third proportional to A and the side of the given square, the rectangle A, EG will be equal to the given square. But the rectangle A, EG, is equal to the rectangles A, HC, and A, CK, i. e. to the rectangles B, IC, and A, CK; .. the rectangles A, KC, and B, IC, are equal to the given square.

(45.) Through a given point, to draw an indefinite line, such, that if lines be drawn from two other given points, and forming given angles with it, the rectangle contained by the segments intercepted between the given point and the two lines so drawn, shall be equal to the square of a given line.

G

B

F

Let A be the given point through which the line is to be drawn; B and C the other given points. Join AB, AC; and on them describe segments of circles ADB, AEC, containing angles equal to the given angles. Draw either diameter AF, on which produced take AG such, that the rectangle FA, AG, may be equal to the given square. Draw GE perpendicular to GF; join EA, and produce it both ways; it is the line required.

Join DF. The angles at G and D being right angles, the triangles AGE, ADF are similar,

:. EA : AG :: FA : AD,

..the rectangle EA, AD is equal to the rectangle FA,

AG, i. e. to the given square; and CE, BD form with ED angles equal to the given angles.

If GE does not meet the circle, the problem is impossible.

(46.) Through a given point between two straight lines containing a given angle, to draw a line such that a perpendicular upon it from the given angle may have a given ratio to a line drawn from one extremity of it, parallel to a line given in position.

Let AB, AC be the lines forming. the given angle BAC, and D a point between them, and AE the line given in position. Draw any line FG parallel to AE, and take AH: FG in

E

F

D
M

B

the given ratio; and with the centre A, and radius AH, describe a circle, to which draw FIK a tangent. Join AI; and through D draw LMN parallel to FK, and LO parallel to FG. LN is the line required.

For AI is perpendicular to FK, and ... AM to LN; and LO is parallel to AE,

and FG LO :: AF : AL :: (AI=) AH : AM, :. AM : LO :: AH: FG, i. e. in the given ratio.

(47.) Through a given point between two indefinite straight lines not parallel to one another, to draw a line which shall be terminated by them, so that the rectangle contained by its segments shall be less than the rectangle contained by the segments of any other line drawn through the same point.

Let AB, AC be the given lines meeting in A. In AC take any point D, and make AE=AD. Join DE; and through I the given point draw FIG

pa

rallel to DE. FIG is the line required.

=

K

F

M

Draw the perpendiculars FO, GO meeting in O. Then since ED is parallel to FG, and the angles AED, ADE are equal, .. AFG and AGF are equal. But AFO AGO, each being a right angle, .. OGF=OFG, and OF=OG; a circle .. described from the centre O, and radius OG, will pass through F, and touch AB, AC in G and F, since the angles at G and F are right angles. Let now any other line HKLM be drawn through I, and terminated by AB, AC. Since all other points in AB but G are without the circle, H is without the circle ; .. HM cuts the circle in K; and for the same reason also in L. Now the rectangle KI, IL is equal to the rectangle GI, IF. But the rectangle KI, IL is less than the rectangle HI, IM; .. the rectangle GI, IF is less than the rectangle HI, IM. In the same manner it may be shewn that the rectangle GI, IF is less than the rectangle contained by the segments of any other line drawn through I, and terminated by AB, AC.

SECT. VI.

(1.) To describe an isosceles triangle on a given finite straight line.

Let AB be the given straight line. Produce it, if necessary; and make AC and BD, each equal to one of the equal sides of the triangle. With A and B as centres, and radii

1

ᎠᎪ BC

AC, BD, describe circles, cutting each other in E; join AE, BE; AEB is the triangle required.

For AE=AC=BD=BE.

(2.) To describe a square which shall be equal to the difference of two squares, whose sides are given.

0 B D

Take a straight line AB terminated at A, and cut off AO equal to a side of the greater, and OB equal to a side of the lesser square. With O as centre, and radius OA, describe a circle OCD; and from B draw BC at right angles to AD. The square described upon BC is the square required.

Join OC. (Eucl. i. 48.), the square described upon BC is equal to the difference of the squares on OC and OB, i. e. on AO and OB.

COR. Hence a mean proportional between the sum and difference of two given lines may be determined.