<Tri-2025-2237>, hint

Let P' on AB, N' on AC be intersections of lines CP, BN with AB, AC respectively.

BM'/CM'=(AB sin M'AB)/(AC sin M'AC)=AB/AC x AG/AH

<HM/GM=1=(AH sin M'AB/AG sin M'AC)..> sin M'AB/ sin M'AC=AG/AH>

Similarly CN'/BN'=BC/AB x BI/BD,   AP'/BP'=AC/BC x CE/CF

Since AH=BI, BD=CE, CF=AG   AM', BN', CP' are concurrent.