Max Incircle Problem

Geometry Level 3

$\triangle ABC$ has a perimeter $3$ . $D$ is on $\overline{AB}$ ,and $E$ is on $\overline{AC}$

$\overline{DE}$ is tangent to the incircle of $\triangle ABC$ and it is parallel to $\overline{BC}$ ,then what is the maximum value of $\overline{DE}?$

The answer is 0.375.

1 solution

Maria Kozlowska
Jul 29, 2018

Let $BC=a$ and height from A to BC be $h$ .

Inradius $r=\dfrac{\triangle ABC}{1.5}=\dfrac{a h}{3}$ .

$\triangle ABC$ is similar to $\triangle ADE$ .

$\dfrac{a}{h}=\dfrac{DE}{h-2r} \Rightarrow DE=\dfrac{a(3-2a)}{3}$

Finding a maximum for a quadratic function we get the value of $\dfrac{3}{8}=\boxed{0.375}$

Max Incircle Problem

The answer is 0.375.

1 solution

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