The Mathematics Education Department
EMT 725, Lisa F. Garey

Inequality in a Triangle

Click here for the Problem statement

Following the suggestions given in the hint, I used differential calculus to maximize f(x)=(1-x)(1+x)(1+x).

Simplifying,

Differentiating, . Setting the derivative to zero yields

(-3x+1)(x+1)=0, so x=1/3 or -1 (inadmissable). Hence the maximum of (PA)(PB)(PC)=f(1/3)=1.185 ocurs when the point P is at 1/3 the radius of the circle from the center.

Click here for an active GSP sketch to see how the product varies as P moves along BC.

Now we can construct a triangle inscribed in a circle that maximizes (PA)(PB)(PC). The construction begins with x being 1/3 of the radius of the circle. We also know that vertex A lies on the diameter that passes through P. The GSP script provides a step-by-step construction and another GSP sketch displays the result.

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