Heights Spiral

In the triangle ABC with sides a=BC, b=CA, c=AB,
draw the infinite clock-wise spiral of heights:
h1=CD ( CD _|_ AB),
h2=DE ( DE _|_ BC),
h3=EF ( EF _|_ CD),
h4=FG ( FG _|_ DE),
h5=GH ( GH _|_ EF), etc.

1. what is the length of the spiral, i.e. the infinite sum h1+h2+h3+...?
2. what are coordinates of the limiting point of the spiral?
3. consider some particular cases:
a=b=c (the regular triangle),
a=3,b=4,c=5 (the smallest Pythagorean triangle), etc.
4. what can be done at the general case of the arbitrary triangle ABC?