## Subject 1.

Consider the triangle $ABC$ where $m(\hat{C})=2 \cdot m(\hat{A})$ and $AC=2 \cdot BC$. Determine the measures of the angles of triangle $ABC$.

Collection: Gazeta Matematică, series B

| Solution Details | Associated Scoring |
| :--- | :---: |
| Consider the angle bisector $(CD, D \in AB)$ of angle $\hat{C}$ and the point $M$, the midpoint of segment $[AC]$. Triangle $DAC$ is isosceles with base $[AC]$, so $DM \perp AC$. | 3p |
| Triangles $DMC$ and $DBC$ are congruent (S.A.S.), so $m(\widehat{DBC})=m(\widehat{DMC})=90^{\circ}$. | **3p** |
| We obtain $m(\widehat{A})=30^{\circ}$ and $m(\widehat{C})=60^{\circ}$. | **1p** |
1. Consider triangle $ABC$ where $m(\hat{C})=2 \cdot m(\hat{A})$ and $AC=2 \cdot BC$. Determine the measures of the angles of triangle $ABC$.