A426. A Demonstration of 
Let fall the perpendicular sw, rv, then 
Fv: Fq:: ar: 3q:: (by the last) oq : as 
::Fq: Fw; therefore the rectangle vVFw 
14 =Fq’*. Which is the fourteenth Propo- 
sition. . 
Through the points H and T describe 
a circle cutting HF, TF produced in x 
and y, then the angle DIF = DHT = 
xyF'; consequently xy is parallel to DT. 
If therefore xH be any chord in» a circle, 
and yT another, cutting the former in F, 
xy being joined, from T draw TD pa- 
rallel to xy to meet xH in D, then the 
16 rectangle HFD = FT’. Which is the 
sixteenth Proposition. . 
Prop. XVIII. Let ABC be a triangle inscribed in a 
circle, whose sides AB and AC are equal, and from A any 
line be drawn meeting the circle again in D and BC in E; 
I say that the rectangle DAE is equal to the square of AB. 
Prop. XIX. Things remaining as in the last propo- 
sition, if lines touching the circle in A and C be drawn to 
meet in F, and FD be drawn cutting BC in G; I say that 
the rectangle BCG is equal to the square of CE, 
Prop. XX. Let ABC be a triangle inscribed in a circle 
whose sides AB and AC are equal, and let AD be parallel 
to BC, and taking any point therein D, let the rectangle 
under AD and P be equal to the square of ABor AC, and 
from the points A and D let the lines AE, DE be inflected 
to any point E in the circle, meeting BC in F and G; I 
say the rectangle under FG and P = the rectangle BFC. . 
