Problem
GEO-B3-M06-P006 Corrected Trig Ceva Check
In a triangle, three cevians form angle pairs \(10^\circ,70^\circ\), \(30^\circ,20^\circ\), \(40^\circ,10^\circ\) at the vertices. Prove that the cevians are concurrent.
C. Hint 1. Apply trig Ceva.
D. Hint 2. Prove \(\sin30^\circ\sin40^\circ=\sin20^\circ\sin70^\circ\).
By trig Ceva, it is enough to prove \[ \frac{\sin10^\circ}{\sin70^\circ}\cdot \frac{\sin30^\circ}{\sin20^\circ}\cdot \frac{\sin40^\circ}{\sin10^\circ}=1. \] Cancel \(\sin10^\circ\). It remains to prove \(\sin30^\circ\sin40^\circ=\sin20^\circ\sin70^\circ\).
Since \(\sin30^\circ=\frac12\), \(\sin70^\circ=\cos20^\circ\), the right side is \(\sin20^\circ\cos20^\circ=\frac12\sin40^\circ\), equal to the left side.
A good follow-up to the previous problem: students see which numbers actually work.