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A semicircle is drawn on the side $AB$ of a rectangle $ABCD$.

Let $P$ be a point on the semicircle, and let $X$ and $Y$ be defined as shown in …

An octagon with all equal sides is formed by cutting four congruent right triangles from each corner of the above rectangle with side lengths $w$ and $w - 2$.

Find …

Consider a right triangle with side lengths $x$, $y$, and $z$, and area $A$. Given that $\dfrac {x^4+y^4+z^4}8 =64-A^2$ and that $x+y+z=4A$, find the perimeter of the triangle.

Given that hyperbola $x^2-\dfrac{y^2}{3}=1$ has left focus $F_1$ and right focus $F_2$. Point $P$ on the hyperbola is such that $\angle F_{1}PF_{2}=\dfrac{2 \pi}3$. The angle bisector of $\angle F_{1}PF_{2}$ intersects …

$\widehat{AC}$ and $\widehat{BC}$ are arcs with centers $B$ and $A$ respectively. The circle in the figure passes through the midpoint of $AB$ and touches both the arcs. If $AB=12$, find …

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