[pawel]

If in a triangle sinA,sinB,sinC are in Arithmetic progression,then prove that altitudes are in harmonic progression.

[terryg752]

If in a triangle sinA,sinB,sinC are in Arithmetic progression,then prove that altitudes are in harmonic progression.

Altitudes are: c Sin A, a Sin B, b Sin C

(where c is side opposite angle C, i.e. AB, etc.)

Divide all 3 altitudes by Sin A SinB Sin C , remember a/Sin A = b/Sin B = c/Sin C = say k

You get (c/SinC) / Sin B, (a/Sin A)/Sin C, (b/Sin B) / Sin A

which are: k/Sin B, k/Sin C, k/SinA

Hence the result!

[terryg752]

Pawel, yhy do you disagree? I thought the following is obvious:

if x, y, z are in Arithmetic Progression, then
1/x, 1/y, 1/z are in Harmonic Progression.

I have proved : altitudes are k/Sin B, k/Sin C, k/SinA, where k is a constant.

Since Sin A, Sin B, Sin C are in Arithmetic Progression,
Sin A/k, Sin B/k, Sin C/k are in Arithmetic Progression

Hence k/Sin A, k/Sin B, k/Sin C are in Harmonic Progression.