From Gogeometry
Using :
- In ΔDMF : DM^2=DF^2+FM^2
- (1) DM^2= (r-x/2)^2+x^2
- In ΔAMD : AM^2=AD^2+DM^2 -2xAD (Proposition 13 Euclide Book II)
- r^2= r^2+ DM^2 -2xr
- (2) DM^2=2xr
- (1) and (2) 2xr=(r-x/2)^2+x^2
- 2xr=r^2-xr+x^2/4+x^2
- 3xr=r^2+5x^2/4
- Dividing by x^2 (x not equal to 0) =>3r/x=5/4+(r/x)^2
- Let y=r/x : y^2-3y+5/4=0
- Delta = 4 and y=5/2 or y=1/2
Therefore x=2r/5 or x=2r
- Since the only definition of M and H are
- M belongs to the square and circle A,
- and H belongs to the square and Circle B
- Therefore, there are 2 different solutions:
- x=2r/5 (blue square)
- or x=2r (green square)
