MathLabs

Problem 1

Let PP be a point inside a given triangle ABCABC, and let DD, EE, FF be the feet of the perpendiculars from PP to the lines BCBC, CACA, ABAB respectively. Find all positions of PP for which BCPD+CAPE+ABPF\dfrac{BC}{PD} + \dfrac{CA}{PE} + \dfrac{AB}{PF} is least.
Step 2 of 5: Record the constant sum
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BC⋅PD+CA⋅PE+AB⋅PF=2SABC(constant)BC\cdot PD + CA\cdot PE + AB\cdot PF = 2S_{ABC}\quad(\text{constant})
Detailed analysis

Step 1 shows the weighted sum BC⋅PD+CA⋅PE+AB⋅PFBC\cdot PD + CA\cdot PE + AB\cdot PF never changes as PP moves inside the triangle; it always equals 2SABC2S_{ABC}.