<Tri-2026-2305>, hint

DE//BC, DE=1/2 BC. Since triangle ADN, 30-60-90 AN=2AD=AC

Similarly AM=2AE=AB. Hence triangles ABC and AMN are congruent MN=BC

Triangle EMN is right triangle and L is the midpoint of MN Hence EL=LN=LM=1/2MN

=1/2BC. ED=EL =1/2BC. finally <EDL=<ELD

<Tri-2026-2305>, general, midpoint

<Singapore MO 2006>

 

<Tri-2026-2304>, hint

Since PQ//BC, GQ=2/3DC=1/3 BC, GE/EC=1/3, Similarly GF/FB=1/3

Hence EF//BC. Since triangles EGQ and ECB are similar DE cuts GQ at M, midpoint of GQ

GQ=1/6BC GD=1/3AD while BD=1/2BC, triangles DGM and ADBare similar, AB//MD AB//ED

Similarly AC//FD. Finally triangles ABC and DEF are similar.

<Tri-2026-2304>, general, similar 

<Singapore MO 2005>

<Solid-2026-270>, hint