Convert intersection of two lines into an arc

Question

I am struggling with finding coordinates of a point for arc origin.

I am programming a tool which converts a corner (intersection of two lines) into an arc of needed radius.

SETUP PICTURE

I have intersecting line segments AD and AC.

Known:

• Coordinates of Point A
• Length of Segment BE (r)
• Angle between AC and AB (which also equals to angle between AD & AB)

My problem is finding any dimension related to segment AB (length, projections, etc), which if known, I can solve the problem.

Ultimately Need:

• Points D, B, and C to construct an Arc and resize the intersecting lines to end at respective points C & D.

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Answer 1

$\triangle{ACE}$ is a right triangle, therefore $${r\over AE} = \sin{\angle{CAE}} = \sin{\left(\frac12\angle{CAD}\right)}.$$ This lets you find $E$, and from there $B$, $C$ and $D$ are easily found. Note,too, that $\angle{AEC}=\frac\pi2-\angle{CAE}$.