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Learn about packing shapes neatly into others, and the resulting geometric properties.

In the above diagram, \(\triangle ABC\) is an equilateral triangle and point \(O\) is the center of the circle inscribed in \(\triangle ABC.\) If the length of \(\overline{AB}\) is \(\lvert \overline{AB} \rvert = 27,\) \(\overline{AB} \text{ and } \overline{OD}\) are parallel, and \(\overline{AC} \text{ and } \overline{OE}\) are parallel, what is \(\lvert\overline{DE}\rvert(=x)?\)

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In the above diagram, circle \(O\) is the incircle of right triangle \(ABC.\) Given the two side lengths \[\lvert\overline{AB}\rvert = 7 \text{ and }\lvert\overline{BC}\rvert = 16,\] what is the area of the circle?

**Note:** The above diagram is not drawn to scale.

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In the above diagram, circle \(O\) is inscribed in triangle \(\triangle ABC,\) with points of contact \(D, E,\) and \(F\). If \(\angle DBE = 54^{\circ} \text{ and } \angle DAF = 46^{\circ},\) what is \(\angle BOC?\)

**Note:** The above diagram is not drawn to scale.

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In the above diagram, circle \(O\) is inscribed in \(\triangle ABC\), with points of contact at points \(D, E\) and \(F\). If \(a=4, b=7\) and \(c=9,\) what is the perimeter of triangle \(\triangle ABC?\)

**Note:** The above diagram is not drawn to scale.

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In the above diagram, circle \(O\) with radius \(r=20\) is inscribed in \(\triangle ABC\). If the perimeter of \(\triangle ABC\) is \(x\) and the area of \(\triangle ABC\) is \(y\), what is \(\frac{y}{x}?\)

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