Problem
GEO-B3-M06-P005 Checking the Trig Ceva Condition
In triangle \(ABC\), cevians \(AA_1,BB_1,CC_1\) satisfy \(\angle BAA_1=20^\circ\), \(\angle CAA_1=40^\circ\), \(\angle CBB_1=30^\circ\), \(\angle ABB_1=50^\circ\), \(\angle ACC_1=40^\circ\), \(\angle BCC_1=30^\circ\). Check whether concurrence of the cevians follows from these data.
C. Hint 1. Write the trig Ceva product.
D. Hint 2. Use \(\sin40^\circ=\sin140^\circ\) and \(\sin50^\circ=\cos40^\circ\).
By trig Ceva, one must check \[ \frac{\sin20^\circ}{\sin40^\circ}\cdot\frac{\sin30^\circ}{\sin50^\circ}\cdot\frac{\sin40^\circ}{\sin30^\circ}=1. \] After canceling \(\sin40^\circ\) and \(\sin30^\circ\), this becomes \(\frac{\sin20^\circ}{\sin50^\circ}\).
Since \(\sin20^\circ\ne\sin50^\circ\), the trig Ceva condition is not satisfied. Therefore concurrence of the cevians does not follow from these data.
This problem deliberately teaches not to trust nice-looking numbers without checking.