How do I find the equation of a curve formed by a series of lines tangent to a curve?

The curve you have described is in fact a parabola; specifically, it is the parabola generated as the quadratic Bézier curve with the control points as the ends of your lines.

The equation of this parabola is, for three points $a, b, c$:

$$f(t)=(1-t)^2a+t(1-t)b+t^2c$$

This is a parametric formula; you enter a $t$ value and it gives the $x$ and $y$ values of a point on the curve, as opposed to entering an $x$ value and getting the $y$ value out.

In the first case, with the right angle, the parabola is simply

$$f(t)=\left(10t^2,10(1-t)^2\right)$$

shown in red on this figure.

The $60^\circ$ angle looks like this.