Školjka

%V0 In an acute-angled triangle $ABC,$ let $AD,BE$ be altitudes and $AP,BQ$ internal bisectors. Denote by $I$ and $O$ the incenter and the circumcentre of the triangle, respectively. Prove that the points $D, E,$ and $I$ are collinear if and only if the points $P, Q,$ and $O$ are collinear.

%V0 For a triangle $ABC,$ let $k$ be its circumcircle with radius $r.$ The bisectors of the inner angles $A, B,$ and $C$ of the triangle intersect respectively the circle $k$ again at points $A', B',$ and $C'.$ Prove the inequality $$16Q^3 \geq 27 r^4 P,$$ where $Q$ and $P$ are the areas of the triangles $A'B'C'$ and $ABC$ respectively.

%V0 Let $ABC$ be an acute-angled triangle. Let $L$ be any line in the plane of the triangle $ABC$. Denote by $u$, $v$, $w$ the lengths of the perpendiculars to $L$ from $A$, $B$, $C$ respectively. Prove the inequality $u^2\cdot\tan A + v^2\cdot\tan B + w^2\cdot\tan C\geq 2\cdot S$, where $S$ is the area of the triangle $ABC$. Determine the lines $L$ for which equality holds.

%V0 Let $ABC$ be an arbitrary triangle and $M$ a point inside it. Let $d_a, d_b, d_c$ be the distances from $M$ to sides $BC,CA,AB$; $a, b, c$ the lengths of the sides respectively, and $S$ the area of the triangle $ABC$. Prove the inequality $$abd_ad_b + bcd_bd_c + cad_cd_a \leq \frac{4S^2}{3}.$$ Prove that the left-hand side attains its maximum when $M$ is the centroid of the triangle.

%V0 Suppose medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that: a.) Using medians of that triangle it is possible to construct a rectangular triangle. b.) The following inequality: $$5(a^2+b^2-c^2) \geq 8ab,$$ is valid, where $a,b$ and $c$ are side length of the given triangle.

%V0 In a given right triangle $ABC$, the hypotenuse $BC$, of length $a$, is divided into $n$ equal parts ($n$ and odd integer). Let $\alpha$ be the acute angel subtending, from $A$, that segment which contains the mdipoint of the hypotenuse. Let $h$ be the length of the altitude to the hypotenuse fo the triangle. Prove that: $$\tan{\alpha}=\dfrac{4nh}{(n^2-1)a}.$$