Insert an arc with a specific radius at point of intersection of 2 lines

Hi, I have a path which is represented by 2 straight lines connected together with a sharp turn. I want to replace this turn by inserting an arc. Currently I am trying to fit a circle with a fixed radius at the point of intersection as shown in the image :

From this I will then find the points where the circle intersects the 2 lines such that the new path can be obtained. However, I am having some trouble to calculate the centre point of the circle. Currently I am considering the test case when the first line is a straight line and the second line makes an angle greater than 90 with the first line are. To obtain the required results the following center point equation are used:

    theta_dif = theta - 90;
    center_x = x2 - r+r*sind(theta_dif);
    center_y = y2 - r;
where r is the radius , theta is the angle between the 2 lines

These equations only hold when the first line has a constant gradient of 0. I need to come up with a generic solution that holds for all angles and line orientation.

Please help. Can I implement it in a different method?

Thanks