Fie \( ABC \) un triunghi cu \( m(\widehat{C})=60\textdegree \), \( L \) proiectia lui \( A \) pe \( BC \), \( M \) proiectia lui \( B \) pe \( AC \), iar \( D \) mijlocul lui \( [AB] \). Demonstrati ca triunghiul \( DML \) este echilateral.
Neculai Roman, Mircesti (Iasi), Recreatii Matematice 1/2009
triunghi echilateral
Moderators: Bogdan Posa, Laurian Filip
-
Andi Brojbeanu
- Bernoulli
- Posts: 294
- Joined: Sun Mar 22, 2009 6:31 pm
- Location: Targoviste (Dambovita)
-
Mateescu Constantin
- Newton
- Posts: 307
- Joined: Tue Apr 21, 2009 8:17 am
- Location: Pitesti
Avem 2 cazuri:
1) \( \triangle ABC \) ascutitunghic
Din \( \triangle AMB \) si \( \triangle ALB \) \( \Longrightarrow MD=LD=\frac{AB}{2} \ (*) \)
\( m(\angle C)=60^{\circ} \Longrightarrow m(\angle CAL)=30^{\circ} \)
\( \Longrightarrow m(\angle ALD)=m(\angle LAD)=m(\angle A)-30^{\circ} \ (1) \) (\( \triangle ALD \) este isoscel)
Patrulaterul \( AMLB \) este inscriptibil, deoarece \( m(\angle AMB)=m(\angle ALB)=90^{\circ} \)
\( \Longrightarrow m(\angle ALM)=m(\angle ABM)=m(\angle B)-30^{\circ} \ (2) \)
Din \( (1),\ (2) \) \( \Longrightarrow m(\angle MLD)=m(\angle A)+m(\angle B)-60^{\circ}=60^{\circ} \ (**) \)
Din \( (*),\ (**) \) rezulta ca \( \triangle MLD \) este echilateral.
2) \( \triangle ABC \) obtuzunghic (Fie \( \angle A \) obtuz)
La fel \( MD=LD \) si \( m(\angle MLD)=90^{\circ}-m(\angle ALM)-m(\angle BLD)=90^{\circ}-[30^{\circ}-m(\angle B)]-m(\angle B)=60^{\circ} \), iar concluzia se impune.
1) \( \triangle ABC \) ascutitunghic
Din \( \triangle AMB \) si \( \triangle ALB \) \( \Longrightarrow MD=LD=\frac{AB}{2} \ (*) \)
\( m(\angle C)=60^{\circ} \Longrightarrow m(\angle CAL)=30^{\circ} \)
\( \Longrightarrow m(\angle ALD)=m(\angle LAD)=m(\angle A)-30^{\circ} \ (1) \) (\( \triangle ALD \) este isoscel)
Patrulaterul \( AMLB \) este inscriptibil, deoarece \( m(\angle AMB)=m(\angle ALB)=90^{\circ} \)
\( \Longrightarrow m(\angle ALM)=m(\angle ABM)=m(\angle B)-30^{\circ} \ (2) \)
Din \( (1),\ (2) \) \( \Longrightarrow m(\angle MLD)=m(\angle A)+m(\angle B)-60^{\circ}=60^{\circ} \ (**) \)
Din \( (*),\ (**) \) rezulta ca \( \triangle MLD \) este echilateral.
2) \( \triangle ABC \) obtuzunghic (Fie \( \angle A \) obtuz)
La fel \( MD=LD \) si \( m(\angle MLD)=90^{\circ}-m(\angle ALM)-m(\angle BLD)=90^{\circ}-[30^{\circ}-m(\angle B)]-m(\angle B)=60^{\circ} \), iar concluzia se impune.