How can I fit a rhombus optimally into a generic triangle and calculate its sidelength I have the following task: calculate largest possible side length of a rhombus inside a generic triangle with side lengths $a, b, c$. And all points of the rhombus need to be on the sides of the triangle. Also one point of the rhomboid is on the point C of the triangle I have no idea how to even sketch that, let alone calculate its length. Can somebody tell me how to start? How to optimally fit a rhombus inside a triangle? I have no idea how I should look at this problem.

Let D be the point on AB such that CD is the angle bisector of $\angle ACB$.

Let the perpendicular bisector of CD cut CA and CB at M and N.

CMDN is the **only** rhombus that meets the requirement can be constructed.