Junction between curved and straight edge in KiCAD
It seems that it is not possible to snap to the circle. Eagle has the same difficulty. However the problem can be solved using some relatively trivial maths. Its basically a geometry problem of trying to find the coordinate of where the line intersects the circle, and then setting the XY coordinate of the end of the line equal to that point.

A diagram will help:



You know both the origin coordinates of the circle \$(X_0,Y_0)\$ and also where the line is on the x-axis - the \$X_1\$ part of the coordinate. Finally you also know the radius \$r\$ of the circle. That is all the information required to find \$Y1\$.

From the diagram, we can say that: 
$$\mathrm{d}X = X_0 - X_1 \tag 1$$

From there we can simply use Pythagoras's Theorem:

$$\mathrm{d}Y = \sqrt{r^2 - \mathrm{d}X^2} \tag 2$$

Then:

$$Y_1 = Y_0 + \mathrm{d}Y \tag 3$$

Putting (1) and (2) into (3) it becomes:

$$Y_1 = Y_0 + \sqrt{r^2 - (X_0 - X_1)^2} \tag 4$$

If you solve (4) for each of your two lines, that will give you the y-axis position for the end of each of the lines such that the line ends on the circle.

In Eagle I would then draw an arc which ends on both of those two points. You can work out the total angle of the arc by using trigonometry to get \$\theta\$ for each of the two lines and find that \$\theta_{arc} = 360^\circ - (\theta_1 - \theta_2)\$. That draws an arc that is the correct radius and intersects with the two lines. I assume you can do the same thing in KiCAD.