282 Investigation of the Epicycloid. 
_ 6. As BD is perpendicular to BG, if the radius DK be drawn 
perpendicular to 1B or GH, the angle BDK will be equal to the an- 
gle BGH; and farther, as BD and BG are equal by construction, 
and likewise as BI or GH equals AD or DK by construction, the 
two triangles BDK and BGH are identical, and BK equals BH. 
But since BH represents the motion of the point B in the epi- 
eycloid, its equal BK may be taken as the measure of the same mo- 
tion, and as the center C of the generating circle has but one motion, 
viz. the uniform one about the center D, represented by CD, it fol- 
lows, that the motion of the center C: is to that of the point B as 
CD to BK. Moreover, as DK and CB are both perpendicular to 
- the same line IB, they are parallel to each other, and as the points 
A, C and D, are in the same right line, the angles ACB and ADK 
are equal; hence the two isosceles triangles ADK and ACB are 
similar and have the vertex A common, and CD : BK: *AC.; AB. 
But, as has been shown, CD is to BK as the motion of the center C 
of the generating circle to that of the point B in the epicycloid ; con- 
sequently, AC is to AB as the same two motions; or, the motion of 
the center C of the generating circle is to that of any point B in tts 
circumference, as the radius of the same circle to the chord drawn from 
the said point B to the point of contact of the two circles at A. 
7. The triangles GBH and DBK being identical, and BG in the 
one being perpendicular to the corresponding side BD in the other, 
the side BH is likewise perpendicular to the corresponding side BK; 
and.as ABF is a semicircle, BH produced will cut the extremity F 
of the diameter AF, and as BH is a tangent to the epicycloid at the 
point B, it follows, that the chord BF in the generating circle 1s par 
allel to the tangent drawn to the point B in the epicycloid. 
8. When by the revolution of the generating circle the point B ar- 
rives at F, the chord AB, which represents the velocity or motion 0 
B, will become equal to the diameter AF or double the radius AC. 
Hence, the motion of the point F is double that of the center C. 
In like manner, when the point B arrives at A, the chord AB 
vanish, and consequently, the point A is at rest. 
9: Since the chords of equal ares have the same ratio to the 1° 
of their respective circles, it follows, (6.) that the relative velocities 
or motions of the center and generating point of the same circle ™ 
describing corresponding portions of the epicycloidal line 
always the same, and consequently, 
the radii 
remains 
