Question

For a given triangle ABC, where \(h_X\) stands for altitude from a vertex \(X\), is \(h_b+h_c>b\) for all \(b\)?

Answer 1

\[h_c = a \sin B\]\[h_b = c \sin A\]\[b^2 = a^2+c^2-2a c\cos B\]\[b^2\sin^2 A \sin^2 B=h_c^2\sin^2 A+h_b^2\sin^2B-2h_ch_b \cos B \sin B\sin A\]