Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points in the interiors of the sides $ BC$, $ CA$, $ AB$ of this triangle. Prove that the area of at least one of the three triangles $ AQR$, $ BRP$, $ CPQ$ is less than or equal to one quarter of the area of triangle $ ABC$. [i]Alternative formulation:[/i] Let $ ABC$ be a triangle, and let $ P$, $ Q$, $ R$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Prove that $ \min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|$, where the abbreviation $ \left|P_1P_2P_3\right|$ denotes the (non-directed) area of an arbitrary triangle $ P_1P_2P_3$.

Let $ABC$ be a right triangle with $AB = BC = 2$. Let $ACD$ be a right triangle with angle $\angle DAC = 30$ degrees and $\angle DCA = 60$ degrees. Given that $ABC$ and $ACD$ do not overlap, what is the area of triangle $BCD$?

Three lines $k, l, m$ are drawn through a point $P$ inside a triangle $ABC$ such that $k$ meets $AB$ at $A_1$ and $AC$ at $A_2 \ne A_1$ and $PA_1 = PA_2$, $l $ meets $BC$ at $B_1$ and $BA$ at $B_2 \ne B_1$ and $PB_1 = PB_2$, $m$ meets $CA$ at $C_1$ and $CB$ at $C_2\ne C_1$ and $PC_1=PC_2$. Prove that the lines $k,l,m$ are uniquely determined by these conditions. Find point $P$ for which the triangles $AA_1A_2, BB_1B_2, CC_1C_2$ have the same area and show that this point is unique.

Prove that for every tetrahedron there exist two planes such that the projection areas on those planes ratio is not less than $\sqrt 2$.

Let $ABC$ be a triangle and $O$ be its circumcircle. Let $A', B', C'$ be the midpoints of minor arcs $AB$, $BC$ and $CA$ respectively. Let $I$ be the center of incircle of $ABC$. If $AB = 13$, $BC = 14$ and $AC = 15$, what is the area of the hexagon $AA'BB'CC'$? Suppose $m \angle BAC = \alpha$ , $m \angle CBA = \beta$, and $m \angle ACB = \gamma$. [b]p10.[/b] Let the incircle of $ABC$ be tangent to $AB, BC$, and $AC$ at $J, K, L$, respectively. Compute the angles of triangles $JKL$ and $A'B'C'$ in terms of $\alpha$, $\beta$, and $\gamma$, and conclude that these two triangles are similar. [b]p11.[/b] Show that triangle $AA'C'$ is congruent to triangle $IA'C'$. Show that $AA'BB'CC'$ has twice the area of $A'B'C'$. [b]p12.[/b] Let $r = JL/A'C'$ and the area of triangle $JKL$ be $S$. Using the previous parts, determine the area of hexagon $AA'BB'CC'$ in terms of $ r$ and $S$. [b]p13.[/b] Given that the circumradius of triangle $ABC$ is $65/8$ and that $S = 1344/65$, compute $ r$ and the exact value of the area of hexagon $AA'BB'CC'$. PS. You had better use hide for answers.

Let $ABCD$ be a convex quadrilateral and $S$ an arbitrary point in its interior. Let also $E$ be the symmetric point of $S$ with respect to the midpoint $K$ of the side $AB$ and let $Z$ be the symmetric point of $S$ with respect to the midpoint $L$ of the side $CD$. Prove that $(AECZ) = (EBZD) = (ABCD)$.

$ABCD$ is a quadrilateral, $E, F, G, H$ are the midpoints of $AB$, $BC$, $CD$, $DA$ respectively. Find the point $P$ such that area $(PHAE) =$ area $(PEBF) =$ area $(PFCG) =$ area $(PGDH).$

Let $A', B', C', D', E'$ and $F'$ be the midpoints of the sides $AB$, $BC$, $CD$, $DE$, $EF$ and $FA$ of an arbitrary convex hexagon $ABCDEF$ respectively. Express the area of $ABCDEF$ in terms of the areas of the triangles $ABC$, $BCD'$, $CDS'$, $DEF'$, $EFA'$ and $FAB'$. (A Lopshi tz, NB Vassiliev)

Sides $AB$ and $CD$ of quadrilateral $ABCD$ intersect at point $E$. On the diagonals$ AC$ and $BD$ points $M$ and $N$ are taken, respectively, so that $AM / AC = BN / BD = k$. Find the area of ​​a triangle $EMN$ if the area of ​​$ABCD$ is $S$.

Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.

The points $M$ and $P$ are the midpoints of $[BC]$ and $[CD]$ sides of a convex quadrangle $ABCD$. It is known that $|AM| + |AP| = a$. Prove that $ABCD$ has area less than $\frac{a^2}{2}$.

The width of rectangle $ABCD$ is twice its height, and the height of rectangle $EFCG$ is twice its width. The point $E$ lies on the diagonal $BD$. Which fraction of the area of the big rectangle is that of the small one? [img]https://1.bp.blogspot.com/-aeqefhbBh5E/XzcBjhgg7sI/AAAAAAAAMXM/B0qSgWDBuqc3ysd-mOitP1LarOtBdJJ3gCLcBGAsYHQ/s0/2004%2BMohr%2Bp1.png[/img]

Five different points $O,A,B,C,D$ are given in plane such that \[OA\le OB\le OC\le OD.\] Show that for area $P$ of any convex quadrilateral with vertices $A,B,C,D$ (not necessarily in this order) the inequality \[P\le \frac12(OA+OD)(OB+OC)\] holds and determine when equality occurs.

Two equal rectangles of area $10$ are arranged as follows. Find the area of the gray rectangle. [img]https://cdn.artofproblemsolving.com/attachments/7/1/112b07530a2ef42e5b2cf83a2cb9fb11dfc9e6.png[/img]

In a triangle $ABC$ with $\angle BC A = 90^o$, the perpendicular bisector of $AB$ intersects segments $AB$ and $AC$ at $X$ and $Y$, respectively. If the ratio of the area of quadrilateral $BXYC$ to the area of triangle $ABC$ is $13 : 18$ and $BC = 12$ then what is the length of $AC$?