Practice (48)
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Given $\triangle{ABC}$, let $m_a, m_b,$ and $m_c$ be the lengths of three medians. Find its area $S_{\triangle{ABC}}$ with respect to $m_a, m_b,$ and $m_c$.
In $\triangle ABC$ let $I$ be the center of the inscribed circle, and let the bisector of $\angle ACB$ intersect $\overline{AB}$ at $L$. The line through $C$ and $L$ intersects the circumscribed circle of $\triangle ABC$ at the two points $C$ and $D$. If $LI=2$ and $LD=3$, then $IC= \frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Let $P$ be a point inside a unit square $ABCD$. Find the minimal value of $AP+BP+CP$
Let $H$ be the orthocenter of acute $\triangle{ABC}$. Show that $$a\cdot BH\cdot CH + b\cdot CH\cdot AH+c\cdot AH\cdot BH=abc$$
where $a=BC, b=CA,$ and $c=AB$.
In $\triangle{ABC}$, let $\angle{A}=120^\circ$. If $A'$, $B'$ and $C'$ are feet of the three interior angle bisectors as shown, prove $A'B'\perp A'C'$.
The incircle $\omega$ of triangle $ABC$ is tangent to $\overline{BC}$ at $X$. Let $Y \neq X$ be the other intersection of $\overline{AX}$ with $\omega$. Points $P$ and $Q$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, so that $\overline{PQ}$ is tangent to $\omega$ at $Y$. Assume that $AP = 3$, $PB = 4$, $AC = 8$, and $AQ = \dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Let $O$ and $H$ be the circumcenter and orthocenter of $\triangle{ABC}$ respectively. Show that $OH\parallel BC$ if and only if $\tan{B}\tan{C}=3$.