Your method sounds promising, However, I have to check it thoroughly. ( I mean I want to check if it yields true tangent point) needless to say it sounds promising.

One thing that made me suspicious about it is that when I zoom in I see two different crossing! ( I mean when I zoom in it seems the curve crosses the red line at two different points which means this method does not show the true tangent point)

I checked it using Trim method and it seems your method is correct way of finding the tangent point!