Problem
GEO-B3-M06-P010 Angle Bisectors via Trig Ceva
#10
★★☆☆☆ Level 2 of 5
Use trig Ceva to prove that the internal angle bisectors of triangle \(ABC\) are concurrent.
Inspired by Prasolov trigonometric geometry method
C. Hint 1. For each angle bisector, the two angles are equal.
D. Hint 2. The product of three sine ratios is immediately \(1\).
Let \(AA_1,BB_1,CC_1\) be the internal angle bisectors. Then \(\angle BAA_1=\angle CAA_1\), so the first trig Ceva factor is \(1\). Similarly, the second and third factors are \(1\).
The product is \(1\), hence by trig Ceva the angle bisectors are concurrent.
A training problem for recognising the theorem's form.