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 4 of 5: Divide by the constant area sum
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BCPD+CAPE+ABPF≥(AB+BC+CA)22SABC\frac{BC}{PD}+\frac{CA}{PE}+\frac{AB}{PF} \ge \frac{(AB+BC+CA)^2}{2S_{ABC}}
Detailed analysis

Since the first factor on the left of the Cauchy–Schwarz inequality equals the constant 2SABC2S_{ABC} from Step 2, dividing both sides by it turns the inequality into an explicit, fixed lower bound for the quantity we want to minimize.