<Tri-2025-2236>, hint

In triangles AMN, ADC, A,P,Q are collinear MP=NP, QD=CQ, then MN//DC

Suppose M'PN', while MP=PN, then M'P cannot be equal to PN', hence MN//DC

and  <ADC=<AMN=<ACB, triangles ABC and ACD are similar 

AC/AD=AB/AC--> AC xAC=AB x AD=2 AD xAD