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Scholium. When the point A lies without the circle, two tangents may always be drawn; for the circumference whose center is D intersects the given circumference in two points.
To inscribe a circle in a given triangle.
Let ABC be the given triangle; it is
A required to inscribe a circle in it.
Bisect the angles B and C by the lines BD, CD,
meeting each other in the point D. From the point of inter
E section, let fall the perpendiculars DE, DF, DG on the three sides of the triangle; these perpendiculars will all be equal. For, by construction, the angle
B F EBD is equal to the angle FBD; the right angle DEB is equal to the right angle DFB; hence the third angle BDE is equal to the third angle BDF (Prop. XXVII., Cor. 2, B. I.). Moreover, the side BD is common to the two triangles BDE, BDF, and the angles adjacent to the common side are equal; therefore the two triangles are equal, and DE is equal to DF. For the same reason, DG is equal to DF. Therefore the three straight lines DE, DF, DG are equal to each other; and if a circumference be described from the center D, with a radius equal to DE, it will pass through the extremities of the lines DF, DG. It will also touch the straight lines AB, BC, CA, because the angles at the points E, F, G are right angles (Prop. IX., B. III.). Therefore the circle EFG is inscribed in the triangle ABC (Def. 11, B. III.)
Scholium. The three lines which bisect the angles of a triangle, all meet in the same point, viz., the center of the in scribed circle.
Upon a given straight line, to describe a segment of a circle which shall contain a given angle.
Let AB be the given straight line, upon which it is required to describe a segment of a circle containing a given angle.
At the point A, in the straight line AB, make the angle 3AD equal to the given angle; and from the point A draw
AC perpendicular to AD. Bisect AB in E, and from E draw EC perpendicular to AB. From the point C, where these perpendiculars meet, with a radius equal to AC, describe a circle. Then will AGB be the segment required.
For, since AD is a perpendicular at the extremity of the radius AC, it is a tangent (Prop. IX., B. III.); and the angle BAD is measured by half the arc AFB (Prop. XVI., B III.). Also, the angle AGB, being an inscribed angle, is measured by half the same arc AFB; hence the angle AGB is equal to the angle BAD, which, by construction, is equal to the given angle. Therefore all the angles inscribed in the segment AĞB are equal to the given angle.
Scholium. If the given angle was a right angle, the required segment would be a semicircle, described on AB as a diameter.
To divide a given straight line into any number of equal parts, or into parts proportional to given lines.
First. Let AB be the given straight line which it is proposed to divide into any number of equal parts, as, for example, five. From the point A draw the indefinite
D straight line AC, making any angle with AB. In AC take any point D, À E
B and set off AD five times upon AC. Join BC, and draw DE parallel to it; then is AE the fifth part of AB.
For, since ED is parallel to BC, AE: AB : : AD : AC (Prop. XVI., B. IV.). But AD is the fifth part of AC; therefore AE is the fifth part of AB.
Secondly. Let AB be the given straight line, and AC a divided line; it is required to divide AB similarly to AC. Suppose AC to be divided in the points D and E. Place AB, AC so as to contain any angle; join BC, and through the points D, E draw DF, EG parallel to BC. The line AB wil. be divided into parts proportional to those of AC.
For, because DF and EG are both parallel to CB, we have AD: AF::DE: FG :EC:GB (Prop. XVI., Cor. 2, B. IV.).
To find a fourth proportional to three given lines. From any point A draw two straight lines AD, AE, containing any angle DAE; and make AB, BD, AC respectively equal to the proposed lines. Join B, C; and through D draw DE parallel to BC; then will CE be the fourth
B D portional required. For, because BC is parallel to DE, we have
AB: BD :: AC: CE (Prop. XVI., B. IV.). Cor. In the same manner may be found a third proportional to two given lines A and B; for this will be the same as a fourth proportional to the three lines A, B, B.
To find a mean proportional between two given lines. Let AB, BC be the two given straight lines; it is required to find a mean proportional between them.
Place AB, BC in a straight line ; upon AC describe the semicircle ADC; and from the point B draw BD perpendicular A
B to AC. Then will BD be the mean proportional required.
For the perpendicular BD, lefall from a point in the circumference upon the diameter, is a mean proportional between the two segments of the diameter AB, BC (Prop. XXII., Cor., B. IV.); and these segments are equal to the wo given lines.
To divide a given line into two parts, such that the greater part may be a mean proportional between the whole line and the other part.
Let AB be the given straight line ; it is required to divide it into two parts at the point F, such that AB : AF :: AF: FB.
At the extremity of the line AB, erect the perpendicular BC, and make
А T B it equal to the half of AB. From C as a center, with a radius equal to CB, describe a circle. Draw AC cutting the circumference in D; and make AF equal to AD. The line AB will be divided in the point F in the manner required.
For, since AB is a perpendicular to the radius CB at its extremity, it is a tangent (Prop. IX., B. III.); and if we produce AC to E, we shall have AE: AB :: AB: AD (Prop. XXVIII., B. IV.). Therefore, by division (Prop. VII., B. II.), AE-AB : AB :: AB-AD: AD. But, by construction, AB is equal to DE; and therefore AE-AB is equal to AD or AF, and AB-AD is equal to FB. Hence AF: AB :: FB : AD or AF; and, consequently, by inversion (Prop. V
AB : AF::AF: FB. Scholium. The line AB is said to be divided in extreme and mean ratio. An example of its use may be seen in Prop. V., Book VI.
Through a given point in a given angle, to draw a straight line so that the parts included between the point and the sides of the angle, may be equal.
Let A be the given point, and BCD the given angle; it is required to draw through Á a line BD, so that BA may be equal to AD.
Through the point A draw AE parallel to BC; and take DE equal to CE. Through the points D and A draw the line BAD; it B will be the line required.
For, because AE is parallel to BC, we have (Prop. XVI, B. IV.),
DE: EC:: DA: AB.
To describe a square that shall be equivalent to a given parallelogram, or to a given triangle. First. Let ABDC be the given paral- C
D lelogram, AB its base, and CE its altitude. Find a mean proportional between AB and CE (Prob. XIX.), and represent it by X; the square described on X will be equiva- A E B lent to the given parallelogram ABDC.
For, by construction, AB:X::X:CE; hence X’ is equal to AB X CE (Prop. I., Cor., B. II.). But AB X CE is the measure of the parallelogram ; and X2 is the measure of the square. Therefore the square described on X is equivalent to the given parallelogram ABDC.
Secondly. Let ABC be the given triangle, BC its base, and AD its altitude. Find a mean proportional between BC and the half of AD, and represent it by Y. Then will the square described on Y be equivalent to the triangle ABC.
For, by construction, BC:Y::Y: AD; hence Y' is equivalent to BC X; AD. . But BCX AD is the measure of the triangle ABC; therefore the square described on Y is equivalent to the triangle ABC.
Upon a given line, to construct a rectangle equivalent to a given rectangle. Let AB be the given straight G
F line, and CDFE the given rectangle. It is required to construct on the line AB a rectangle equivalent to CDFE.
D Find a fourth proportional A
B (Prob. XVIII.) to the three lines AB, CD, CE, and let AG be that fourth proportional. The rectangle constructed on che lines AB, AG will be equivale!'t to CDFE.