Why Does My Snell's Law Demonstration Fail Using a Linear Function Approach?

[physics user1]

I can't figure out why my demonstration of snell's law fails, that's the demonstration: (I used a photo)
I think it fails because the function t (HO) represents a line and so the concept of minimum is not defined, when I take the derivative and equal it to 0 I'm considering the case when the line is parallel to the x-axis (the first derivative gives me the angular coefficient and when I equal it to 0 the line is parallel) (considering as y the time and as x HO)

Is that the problem? ( I know the real demonstration but i want to understand why this variant doesn't work)

 

[theodoros.mihos]

$AO\sin\theta_1=AH$

 

[physics user1]

$AO\sin\theta_1=AH$

But... θ1 is the angle whit the vertical

 

[theodoros.mihos]

You are right. May be better a simpler geometry.
Let $A(x_1,y_1)$ and $B(x_2,y_2)$ the end points and $O(x,0)$ the point than we need. Write the total time and take derivation $d/dx(t)=0$.

 

[toumaza]

yep it is preferable to use simple geometry to demonstrate