Detect if circle crossing arc, Lua

How can I detect whether a circle intersects or crosses a given arc segment?

The arc is defined by a center, radius, start angle (angle1), and end angle (angle2). How to detect that arc AB has intersection with the circle D?

I see that the first step is to check if the distance between points C and D is less than the sum of their radii.

But how do I check if the circle touches only the arc part, not the whole circle?

Now my solution doesn't work well:

The name of the problem: circular obstacle pathfinding

Thanks all for the comments!

It's really easier than I think:

-- check if an arc intersects a circle
local function isArcCircleIntersection(arc, circle, rightDir)
    -- arc.x, arc.y, arc.radius is a first circle
    local intersectionPoints = findCircleIntersections(arc, circle) -- returns array of collision points
    for _, point in ipairs(intersectionPoints) do
        local angle = math.atan2(point.y - arc.y, point.x - arc.x)
        local da1 = normalizeAngle(angle - arc.angle1) -- above
        local da2 = normalizeAngle(arc.angle2 - angle) -- below
        if rightDir then -- the other direction
            da1, da2 = da2, da1
        end
        if da1 >= 0 and da2 >= 0 then
            return true -- the given angle was between of two angles
        end
    end
    return false
end

So the arc will be recognized correctly!