<Tri-2024-2142>, hint

Suppose a line through I cuts AB, BC at M' N' such that n<IM'B=<IKL,< IN'B=<ILK.

Then Triangles IAM' and IAK are congruent, AM'=AK

Similarlu triangles ICN' and ICL are congruent  CN'=CL

Ad Triangles IMM' and INN' are congruent. AM'+CN'=AM+CN