Consider a scalene triangle $ ABC $ with $ AB <AC <BC. $ The $ AB $ side mediator cuts the $ B $ side at the $ K $ point and the $ AC $ prolongation at the $ U. $ point. $ AC $ side cuts $ BC $ side at $ O $ point and $ AB $ side extension at $ G$ point. Prove that the $ GOKU $ quad is cyclic, meaning its four vertices are at same circumference

Find a relation between the angles of a triangle such that this could be separated in two isosceles triangles by a line. [i]Dan Brânzei[/i]

The inner bisections of the $ \angle ABC $ and $ \angle ACB $ angles of the $ ABC $ triangle are at $ I $. The $ BI $ parallel line that runs through the point $ A $ finds the $ CI $ line at the point $ D $. The $ CI $ parallel line for $ A $ finds the $ BI $ line at the point $ E $. The lines $ BD $ and $ CE $ are at the point $ F $. Show that $ F, A $, and $ I $ are collinear if and only if $ AB = AC. $

For a rectangle $ABCD$ which is not a square, there is $O$ such that $O$ is on the perpendicular bisector of $BD$ and $O$ is in the interior of $\triangle BCD$. Denote by $E$ and $F$ the second intersections of the circle centered at $O$ passing through $B, D$ and $AB, AD$. $BF$ and $DE$ meets at $G$, and $X, Y, Z$ are the foots of the perpendiculars from $G$ to $AB, BD, DA$. $L, M, N$ are the foots of the perpendiculars from $O$ to $CD, BD, BC$. $XY$ and $ML$ meets at $P$, $YZ$ and $MN$ meets at $Q$. Prove that $BP$ and $DQ$ are parallel.

Let $ABC$ be an acute-angled triangle. The external bisector of $\angle{BAC}$ meets the line $BC$ at point $N$. Let $M$ be the midpoint of $BC$. $P$ and $Q$ are two points on line $AN$ such that, $\angle{PMN}=\angle{MQN}=90^{\circ}$. If $PN=5$ and $BC=3$, then the length of $QA$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are coprime positive integers. What is the value of $(a+b)$?

5. In acute triangle $ABC$, the lines tangent to the circumcircle of $ABC$ at $A$ and $B$ intersect at point $D$. Let $E$ and $F$ be points on $CA$ and $CB$ such that $DECF$ forms a parallelogram. Given that $AB = 20$, $CA=25$ and $\tan C = 4\sqrt{21}/17$, the value of $EF$ may be expressed as $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by winnertakeover and Monkey_king1[/i]

Show that for any vectors $a, b$ in Euclidean space, \[|a \times b|^3 \leq \frac{3 \sqrt 3}{8} |a|^2 |b|^2 |a-b|^2\] Remark. Here $\times$ denotes the vector product.

Let $ABC$ be an acutangle triangle inscribed in a circle $\Gamma$ of center $O$. Let $D$ be the height of the vertex $A$. Let E and F be points over $\Gamma$ such that $AE = AD = AF$. Let $P$ and $Q$ be the intersection points of the $EF $ with sides $AB$ and $AC$ respectively. Let $X$ be the second intersection point of $\Gamma$ with the circle circumscribed to the triangle $AP Q$. Show that the lines $XD$ and $AO $ meet at a point above sobre

Let K be a bounded, d-dimensional convex polyhedron that is not simplex and P is a point on K. Show that if vertices $P_1 , ..., P_k$ are not all on the same face of K, then one of them can be omitted so that the convex hull of the remaining vertices of K still contains P. [hide=note]caratheodory's theorem might be useful. [/hide]

Let $ABCDEF$ be a hexagon inscribed in a circle of radius $r.$ Show that if $AB=CD=EF=r,$ then the midpoints of $BC, DE$ and $FA$ are the vertices of an equilateral triangle.

Show that for any vectors $a, b$ in Euclidean space, \[|a \times b|^3 \leq \frac{3 \sqrt 3}{8} |a|^2 |b|^2 |a-b|^2\] Remark. Here $\times$ denotes the vector product.

a) In a $ XYZ$ triangle, the incircle tangents the $ XY $ and $ XZ $ sides at the $ T $ and $ W $ points, respectively. Prove that: $$ XT = XW = \frac {XY + XZ-YZ} {2} $$ Let $ ABC $ be a triangle and $ D $ is the foot of the relative height next to $ A. $ Are $ I $ and $ J $ the incentives from triangle $ ABD $ and $ ACD $, respectively. The circles of $ ABD $ and $ ACD $ tangency $ AD $ at points $ M $ and $ N $, respectively. Let $ P $ be the tangency point of the $ BC $ circle with the $ AB$ side. The center circle $ A $ and radius $ AP $ intersect the height $ D $ at $ K. $ b) Show that the triangles $ IMK $ and $ KNJ $ are congruent c) Show that the $ IDJK $ quad is inscritibed

Let $ ABC $ be an acutangle triangle and $ D $ any point on the $ BC $ side. Let $ E $ be the symmetrical of $ D $ in $ AC $ and $ F $ is the symmetrical $ D $ relative to $ AB $. $ A $ straight $ ED $ intersects straight $ AB $ at $ G $, while straight $ F D $ intersects the line $ AC $ in $ H $. Prove that the points $ A, E, F, G$ and $ H $ are on the same circumference.

Let $ ABC $ be an acutangle triangle and $ D $ any point on the $ BC $ side. Let $ E $ be the symmetrical of $ D $ in $ AC $ and $ F $ is the symmetrical $ D $ relative to $ AB $. $ A $ straight $ ED $ intersects straight $ AB $ at $ G $, while straight $ F D $ intersects the line $ AC $ in $ H $. Prove that the points $ A, E, F, G$ and $ H $ are on the same circumference.

Let $ABC$ be an acutangle triangle inscribed in a circle $\Gamma$ of center $O$. Let $D$ be the height of the vertex $A$. Let E and F be points over $\Gamma$ such that $AE = AD = AF$. Let $P$ and $Q$ be the intersection points of the $EF $ with sides $AB$ and $AC$ respectively. Let $X$ be the second intersection point of $\Gamma$ with the circle circumscribed to the triangle $AP Q$. Show that the lines $XD$ and $AO $ meet at a point above sobre