The area of the triangle $ABC$ shown in the figure is $1$ unit. Points $D$ and $E$ lie on sides $AC$ and $BC$ respectively, and also are its ''one third'' points closer to $C$. Let $F$ be that $AE$ and $G$ are the midpoints of segment $BD$. What is the area of the marked quadrilateral $ABGF$? [img]https://cdn.artofproblemsolving.com/attachments/4/e/305673f429c86bbc58a8d40272dd6c9a8f0ab2.png[/img]

In the following figure, the largest square is divided into two squares and three rectangles, as shown: The area of each smaller square is equal to $a$ and the area of each small rectangle is equal to $b$. If $a+b=24$ and the root square of $a$ is a natural number, find all possible values for the area of the largest square. [img]https://cdn.artofproblemsolving.com/attachments/f/a/0b424d9c293889b24d9f31b1531bed5081081f.png[/img]

Let $P = (7, 1)$ and let $O = (0, 0)$. (a) If $S$ is a point on the line $y = x$ and $T$ is a point on the horizontal $x$-axis so that $P$ is on the line segment $ST$, determine the minimum possible area of triangle $OST$. (b) If $U$ is a point on the line $y = x$ and $V$ is a point on the horizontal $x$-axis so that $P$ is on the line segment $UV$, determine the minimum possible perimeter of triangle $OUV$.

On the sides $AB$, $BC$, $CD$, $DA$ of the square $ABCD$ points $P, Q, R, T$ are chosen such that $$\frac{AP}{PB}=\frac{BQ}{QC}=\frac{CR}{RD}=\frac{DT}{TA}=\frac12.$$ The segments $AR$, $BT$, $CP$, $DQ$ in the intersection form the quadrilateral $KLMN$ (see figure). [img]https://cdn.artofproblemsolving.com/attachments/f/c/587a2358734c300fe7082c520f90c91f872b49.png[/img] a) Prove that $KLMN$ is a square. b) Find the ratio of the areas of the squares $KLMN$ and $ABCD$. (Alexander Shkolny)

Let $ABCD$ and $A'B'C'D'$ be two squares, both are oriented clockwise. In addition, it is assumed that all points are arranged as shown in the figure.Then it has to be shown that the sum of the areas of the quadrilaterals $ABB'A'$ and $CDD'C'$ equal to the sum of the areas of the quadrilaterals $BCC'B'$ and $DAA'D'$. [img]https://cdn.artofproblemsolving.com/attachments/0/2/6f7f793ded22fe05a7b0a912ef6c4e132f963e.png[/img]

Prove that you can select two adjacent sides of any quadrilateral and supplement them in order to create a parallelogram, the resulting parallelogram contains the original quadrilateral .

Ten coins of $1$ cm radius are placed around a circle as indicated in the figure. Each coin is tangent to the circle and its two neighboring coins. Prove that the sum of the areas of the ten coins is twice the area of the circle. [img]https://cdn.artofproblemsolving.com/attachments/5/e/edf7a7d39d749748f4ae818853cb3f8b2b35b5.gif[/img]

Let's call a convex figure, the boundary of which consists of two segments and an arc of a circle, a mushroom-gon (see fig.). An arbitrary mushroom-gon is given. Use a compass and straightedge to draw a straight line dividing its area in half. [img]https://cdn.artofproblemsolving.com/attachments/d/e/e541a83a7bb31ba14b3637f82e6a6d1ea51e22.png[/img]

In the rectangle $ABCD, BC = 5, EC = 1/3 CD$ and $F$ is the point where $AE$ and $BD$ are cut. The triangle $DFE$ has area $12$ and the triangle $ABF$ has area $27$. Find the area of the quadrilateral $BCEF$ . [img]https://1.bp.blogspot.com/-4w6e729AF9o/XNY9hqHaBaI/AAAAAAAAKL0/eCaNnWmgc7Yj9uV4z29JAvTcWCe21NIMgCK4BGAYYCw/s400/may%2B2011%2Bl1.png[/img]

A finite set of unit circles is given in a plane such that the area of their union $U$ is $S$. Prove that there exists a subset of mutually disjoint circles such that the area of their union is greater that $\frac{2S}{9}.$

In a garden there is an $L$-shaped fence, see figure. You also have at your disposal two finished straight fence sections that are $13$ m and $14$ m long respectively. From point $A$ you want to delimit a part of the garden with an area of at least $200$ m$^2$ . Is it possible to do this? [img]https://1.bp.blogspot.com/-VLWIImY7HBA/X0yZq5BrkTI/AAAAAAAAMbg/8CyP6DzfZTE5iX01Qab3HVrTmaUQ7PvcwCK4BGAYYCw/s400/sweden%2B16p1.png[/img]

What are the possible areas of a hexagon with all angles equal and sides $1, 2, 3, 4, 5$, and $6$, in some order?

The radius of the circumcircle of triangle $\Delta$ is $R$ and the radius of the inscribed circle is $r$. Prove that a circle of radius $R + r$ has an area more than $5$ times the area of triangle $\Delta$.

The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square. [img]https://1.bp.blogspot.com/-xeFvahqPVyM/XzcFfB0-NfI/AAAAAAAAMYA/SV2XU59uBpo_K99ZBY43KSSOKe-veOdFQCLcBGAsYHQ/s0/1998%2BMohr%2Bp3.png[/img]

Seven distinct points are given inside a square with side length $1.$ Together with the square's vertices, they form a set of $11$ points. Consider all triangles with vertices in $M.$ a) Show that at least one of these triangles has an area not exceeding $1\slash 16.$ b) Give an example in which no four of the seven points are on a line and none of the considered triangles has an area of less than $1\slash 16.$

A bowl is formed by attaching four regular hexagons of side 1 to a square of side 1. The edges of adjacent hexagons coincide, as shown in the figure. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl? [asy] size(200); defaultpen(linewidth(0.8)); draw((342,-662) -- (600, -727) -- (757,-619) -- (967,-400) -- (1016,-300) -- (912,-116) -- (651,-46) -- (238,-90) -- (82,-204) -- (184, -388) -- (447,-458) -- (859,-410) -- (1016,-300)); draw((82,-204) -- (133,-490) -- (342, -662)); draw((652,-626) -- (600,-727)); draw((447,-458) -- (652,-626) -- (859,-410)); draw((133,-490) -- (184, -388)); draw((967,-400) -- (912,-116)^^(342,-662) -- (496, -545) -- (757,-619)^^(496, -545) -- (446, -262) -- (238, -90)^^(446, -262) -- (651, -46),linewidth(0.6)+linetype("5 5")+gray(0.4)); [/asy] $\textbf{(A) }6\qquad\textbf{(B) }7\qquad\textbf{(C) }5+2\sqrt{2}\qquad\textbf{(D) }8\qquad\textbf{(E) }9$

Csenge has a yellow and a red foil on her rectangular window which look beautiful in the morning light. Where the two foils overlap, they look orange. The window is $80$ cm tall, $120$ cm wide and its corners are denoted by $A, B, C$ and $D$ in the figure. The two foils are triangular and both have two of their vertices at the two bottom corners of the window, A and $B$. The third vertex of the yellow foil is $S$, the trisecting point of side $DC$ closer to $D$, whereas the third vertex of the red foil is $P$, which is one fourth on the way on segment $SC$, closer to $C$. The red region (i.e. triangle $BPE$) is of area $16$ dm$^2$. What is the total area of the regions not covered by foil? [img]https://cdn.artofproblemsolving.com/attachments/b/c/ea371aeafde6968506da6f3456e88fa0bddc6d.png[/img]

In an acute triangle $ABC$ the angle bisector $AL$, $L \in BC$, intersects its circumcircle at $N$. Let $K$ and $M$ be the projections of $L$ onto sides $AB$ and $AC$. Prove that triangle $ABC$ and quadrilateral $A K N M$ have equal areas.

A parabola has focus $F$ and vertex $V$ , where $VF = 1$0. Let $AB$ be a chord of length $100$ that passes through $F$. Determine the area of $\vartriangle VAB$.

In a triangle $ABC, X$ and $Y$ are points on the segments $AB$ and $AC$, respectively, such that $AX : XB = 1 : 2$ and $AY :YC = 2:1$. If the area of triangle $AXY$ is $10$, then what is the area of triangle $ABC$?

Prove that if all four vertices of a parallelogram are lattice points and there are some other lattice points in or on the parallelogram, then its area exceeds $1$.

Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$

Let $ABCD$ be a quadrilateral with $\angle DAB=60^o$, $\angle ABC=60^o$ and $\angle BCD=120^o$. Diagonals $AC$, $BD$ intersect at point $M$ and $BM=a, MD=2a$. Let $O$ be the midpoint of side $AC$ and draw $OH \perp BD, H \in BD$ and $MN\perp OB, N \in OB$. Prove that i) $HM=MN=\frac{a}{2}$ ii) $AD=DC$ iii) $S_{ABCD}=\frac{9a^2}{2}$

Let $ABCD$ be a rectangle. Let $K,L$ be the midpoints of $BC, AD$ respectively. From point $B$ the perpendicular line on $AK$, intersects $AK$ at point $E$ and $CL$ at point $Z$. a) Prove that the quadrilateral $AKZL$ is an isosceles trapezoid b) Prove that $2S_{ABKZ}=S_{ABCD}$ c) If quadrilateral $ABCD$ is a square of side $a$, calculate the area of the isosceles trapezoid $AKZL$ in terms of side $BC=a$

The figure shows a line intersecting a square lattice. The area of some arising quadrilaterals are also indicated. What is the area of the region with the question mark? [img]https://cdn.artofproblemsolving.com/attachments/0/d/4d5741a63d052e3f6971f87e60ca7df7302fb0.png[/img]