MathLabs

Problem 3

Consider all triangles ABCABC with a fixed base ABAB and whose altitude from CC is a constant hh. For which of these triangles is the product of its three altitudes a maximum?
Step 2 of 5: Reduce to minimizing the two sides from C
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h ha hb maximal⟺AC⋅BC minimalh\,h_a\,h_b\text{ maximal}\Longleftrightarrow AC\cdot BC\text{ minimal}
Detailed analysis

In the identity from the previous step, AB⋅hAB\cdot h is fixed. Thus the product hhahbh h_a h_b is largest exactly when the denominator AC⋅BCAC\cdot BC is smallest.