The square root of a number

In a coordinate system, draw the points H(-1;0) and A(x;0) x being a positive number. Draw the point I, I being the middle of [HA]. Draw the circle with I as its centre and IA as its radius; it intersects the y-axis in C a point with a positive value on the y-axis.

Demonstration :
AI =
AH
  
=
 

CI = IA = OA - OI = 

 

In the OCI triangle, right-angled at O, based on the Pythagorean theorem CI²=

Thus OC ²=

Using the values that we found earlier : OC ²= , thus OC =