Problem
GEO-B3-M06-P017 Diagonals of an 18-Gon
In a regular \(18\)-gon, prove that the three diagonals which, in a suitable triangle, give angle pairs \(10^\circ,70^\circ\), \(30^\circ,20^\circ\), \(40^\circ,10^\circ\), are concurrent.
C. Hint 1. Translate the problem into a triangle with three cevians.
D. Hint 2. Then use exactly the same trig Ceva check for special angles.
Choose a triangle whose sides are three chords of the regular \(18\)-gon so that the given diagonals become cevians. Since the central angle is \(20^\circ\), all angles between chords are half-arc angles, hence multiples of \(10^\circ\).
The angle pairs become \(10^\circ,70^\circ\), \(30^\circ,20^\circ\), \(40^\circ,10^\circ\). By trig Ceva, \[ \frac{\sin10^\circ}{\sin70^\circ}\cdot \frac{\sin30^\circ}{\sin20^\circ}\cdot \frac{\sin40^\circ}{\sin10^\circ}=1. \] This identity follows from \(\sin70^\circ=\cos20^\circ\). Hence the diagonals are concurrent.
This bridges trigonometry and regular polygons.