No need to use coordinates of points just sides are enough.

We do not have to solve for `s` and `ag`, we already have formulas for them.

Why I used `/. Abs -> Identity`? Because in Wolfram they probably do not know that [Heron's formula][1] should be without `Abs`.

`Solve` provides four solutions, two of them evidently negative so we pick up only other two.

Why there are two solutions? Because the triangle sides can be ordered `a, b, c` or `b, a, c`.

 (* s==Area[SSSTriangle[a,b,c]];
 ag==TriangleMeasurement[SSSTriangle[a,b,c],{"InteriorAngle",2}]; *)
 
 Solve[Area[SSSTriangle[a, b, c]] == 
 Area[SSSTriangle[d, b, jlcd]] + Area[SSSTriangle[a, c - d, jlcd]] /. 
 Abs -> Identity, jlcd, {ag, s}]
 %[[{2, 4}]]

---

[![enter image description here][2]][2]


 [1]: https://en.wikipedia.org/wiki/Heron%27s_formula
 [2]: https://i.sstatic.net/fIheBq6t.png