Show that there is a point $P$ inside the quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal area. Show that $P$ must lie on one of the diagonals.

In the right triangle $ABC$, the perpendicular sides $AB$ and $BC$ have lengths $3$ cm and $4$ cm, respectively. Let $M$ be the midpoint of the side $AC$ and let $D$ be a point, distinct from $A$, such that $BM = MD$ and $AB = BD$. a) Prove that $BM$ is perpendicular to $AD$. b) Calculate the area of the quadrilateral $ABDC$.

In triangle $ABC$, side $AC$ is $8$ cm. Two segments are drawn parallel to $AC$ that have their ends on $AB$ and $BC$ and that divide the triangle into three parts of equal area. What is the length of the parallel segment closest to $AC$?

Given triangle $ABC$ with $|AC| > |BC|$. The point $M$ lies on the angle bisector of angle $C$, and $BM$ is perpendicular to the angle bisector. Prove that the area of triangle AMC is half of the area of triangle $ABC$. [img]https://cdn.artofproblemsolving.com/attachments/4/2/1b541b76ec4a9c052b8866acbfea9a0ce04b56.png[/img]

The goal of this problem is to show that the maximum area of a polygon with a fixed number of sides and a fixed perimeter is achieved by a regular polygon. (a) Prove that the polygon with maximum area must be convex. (Hint: If any angle is concave, show that the polygon’s area can be increased.) (b) Prove that if two adjacent sides have different lengths, the area of the polygon can be increased without changing the perimeter. (c) Prove that the polygon with maximum area is equilateral, that is, has all the same side lengths. It is true that when given all four side lengths in order of a quadrilateral, the maximum area is achieved in the unique configuration in which the quadrilateral is cyclic, that is, it can be inscribed in a circle. (d) Prove that in an equilateral polygon, if any two adjacent angles are different then the area of the polygon can be increased without changing the perimeter. (e) Prove that the polygon of maximum area must be equiangular, or have all angles equal. (f ) Prove that the polygon of maximum area is a regular polygon. PS. You had better use hide for answers.

An arbitrary triangle $ABC$ is given and a point $P$ lies on the side $AB$. It is requested to draw through $P$ a line that divides the triangle into two figures of the same area.

Consider the ellipsoid$$\frac{x^2}{a^2}+\frac{y^2}{a^2}+\frac{z^2}{b^2}=1$$($a$ and $b > 0$) and the ellipse $E$ which is the intersection of the ellipsoid with the plane of equation$$mx + ny + pz = 0$$where the point $P = [m, n, p]$ is a random point from the unit sphere $(m^2 + n^2 + p^2 = 1)$. Consider the random variable $A_E$ the area of the ellipse $E$. If the point $P$ is chosen with uniform distribution with respect to the area on the unit sphere, what is the expectation of $A_E$ ?

On a straight line $\ell$ there are four points, $A$, $B$, $C$ and $D$ in that order, such that $AB=BC=CD$. A point $E$ is chosen outside the straight line so that when drawing the segments $EB$ and $EC$, an equilateral triangle $EBC$ is formed . Segments $EA$ and $ED$ are drawn, and a point $F$ is chosen so that when drawing the segments $FA$ and $FE$, an equilateral triangle $FAE$ is formed outside the triangle $EAD$. Finally, the lines $EB$ and $FA$ are drawn , which intersect at the point $G$. If the area of triangle $EBD$ is $10$, calculate the area of triangle $EFG$.

$O$ is the intersection point of the diagonals of the trapezoid $ABCD$. A line passing through $C$ and a point symmetric to $B$ with respect to $O$, intersects the base $AD$ at the point $K$. Prove that $S_{AOK} = S_{AOB} + S_{DOK}$.

The inscribed circle of $\triangle ABC$ touches $BC, AC$ and $AB$ at $D,E$ and $F$ respectively. Denote the perpendicular foots from $F, E$ to $BC$ by $K, L$ respectively. Let the second intersection of these perpendiculars with the incircle be $M, N$ respectively. Show that $\frac{{{S}_{\triangle BMD}}}{{{S}_{\triangle CND}}}=\frac{DK}{DL}$ by Mahdi Etesami Fard

A square table is covered with a square cloth (may be of a different size) without folds and wrinkles. All corners of the table are left uncovered and all four hanging parts are triangular. Given that two adjacent hanging parts are equal prove that two other parts are also equal.

A teacher gives each student in the class the following task in their exercise book . "Take two concentric circles of radius $1$ and $10$ . To the smaller circle produce three tangents whose intersections $A, B$ and $C$ lie in the larger circle . Measure the area $S$ of triangle $ABC$, and areas $S_1 , S_2$ and $S_3$ , the three sector-like regions with vertices at $A, B$ and $C$ (as in the diagram). Find the value of $S_1 +S_2 +S_3 -S$." Prove that each student would obtain the same result . [img]https://1.bp.blogspot.com/-K3kHWWWgxgU/XWHRQ8WqqPI/AAAAAAAAKjE/0iO4-Yz6p9AcM2mklprX_M18xTyg9O81gCK4BGAYYCw/s200/TOT%2B1985%2BAutumn%2BJ3.png[/img] ( A . K . Tolpygo, Kiev)

A triangle is given. Its side a is of length $20$ cm, and its area is $125$ cm$^2$. It is also known that one of the angles lying on side a is twice as large as the other one. We cut the triangle into two parts at the median belonging to side a. Then we move the so-obtained two parts towards each other, such that the two segments of side a remain on the same line (i.e., the line initially occupied by side a). We move the two parts towards each other until we first reach a moment when the common part of the two segments is of length $4$ cm. What is the area of the so-obtained shape in cm$^2$? The so-obtained shape is the union of the two parts, which is a heptagon. [img]https://cdn.artofproblemsolving.com/attachments/3/0/3d45e2df6a0043dfa4fe5ccf64865da8879b42.png[/img]

The figure below shows a $90 \times90$ square with each side divided into three equal segments. Some of the endpoints of these segments are connected by straight lines. Find the area of the shaded region. [asy] import graph; size(6cm); real labelscalefactor = 0.5; pen dotstyle = black; draw((-4,6)--(86,6)--(86,96)--(-4,96)--cycle); filldraw((16,76)--(-4,36)--(32,60)--(56,96)--cycle,grey); filldraw((32,60)--(-4,6)--(50,42)--(86,96)--cycle,grey); filldraw((50,42)--(26,6)--(66,26)--(86,66)--cycle,grey); draw((-4,6)--(26,6)); draw((26,6)--(56,6)); draw((56,6)--(86,6)); draw((-4,6)--(86,6)); draw((86,6)--(86,96)); draw((86,96)--(-4,96)); draw((-4,96)--(-4,6)); draw((26,96)--(-4,36)); draw((56,96)--(-4,6)); draw((86,96)--(26,6)); draw((86,66)--(56,6)); draw((-4,66)--(56,96)); draw((-4,36)--(86,96)); draw((-4,6)--(86,66)); draw((26,6)--(86,36)); draw((16,76)--(-4,36)); draw((-4,36)--(32,60)); draw((32,60)--(56,96)); draw((56,96)--(16,76)); draw((32,60)--(-4,6)); draw((-4,6)--(50,42)); draw((50,42)--(86,96)); draw((86,96)--(32,60)); draw((50,42)--(26,6)); draw((26,6)--(66,26)); draw((66,26)--(86,66)); draw((86,66)--(50,42)); dot((-4,96),dotstyle); dot((26,96),dotstyle); dot((56,96),dotstyle); dot((86,96),dotstyle); dot((-4,6),dotstyle); dot((-4,36),dotstyle); dot((-4,66),dotstyle); dot((27.09,6),dotstyle); dot((56,6),dotstyle); dot((86,36),dotstyle); dot((86,66),dotstyle); dot((86,6),dotstyle); [/asy]

Two unit squares with parallel sides overlap by a rectangle of area $1/8$. Find the extreme values of the distance between the centers of the squares.

A cube whose edge is $1$ is intersected by a plane that does not pass through any of its vertices, and its edges intersect only at points that are the midpoints of these edges. Find the area of the formed section. Consider all possible cases. (Alexander Shkolny)

Let $P,Q,R$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$. Suppose that $BQ$ and $CR$ meet at $A', AP$ and $CR$ meet at $B'$, and $AP$ and $BQ$ meet at $C'$, such that $AB' = B'C', BC' =C'A'$, and $CA'= A'B'$. Compute the ratio of the area of $\triangle PQR$ to the area of $\triangle ABC$.

Give an acute-angled triangle $ABC$ with area $S$, let points $A',B',C'$ be located as follows: $A'$ is the point where altitude from $A$ on $BC$ meets the outwards facing semicirle drawn on $BX$ as diameter.Points $B',C'$ are located similarly. Evaluate the sum $T=($area $\vartriangle BCA')^2+($area $\vartriangle CAB')^2+($area $\vartriangle ABC')^2$.

What maximal area can have a triangle if its sides $a,b,c$ satisfy inequality $0\le a\le 1\le b\le 2\le c\le 3$ ?

Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$ and let $G$ be the point of the segment $AD$ such that $AG = 2GD$. Let $E$ and $F$ be points on the sides $AB$ and $AC$, respectively, such that$ G$ lies on the segment $EF$. Let $M$ and $N$ be points of the segments $AE$ and $AF$, respectively, such that $ME = EB$ and $NF = FC$. a) Prove that the area of the quadrilateral $BMNC$ is equal to four times the area of the triangle $DEF$. b) Prove that the quadrilaterals $MNFE$ and $AMDN$ have the same area.

If all vertices of a triangle on the square grid are grid points, then the triangle is called a [i]lattice[/i] triangle. What is the area of the lattice triangle with (one) of the smallest area, if one grid has area $1$ square unit?

The figure shows a rectangle, its circumscribed circle and four semicircles, which have the rectangle’s sides as diameters. Prove that the combined area of the four dark gray crescentshaped regions is equal to the area of the light gray rectangle. [img]https://1.bp.blogspot.com/-gojv6KfBC9I/XzT9ZMKrIeI/AAAAAAAAMVU/NB-vUldjULI7jvqiFWmBC_Sd8QFtwrc7wCLcBGAsYHQ/s0/2013%2BMohr%2Bp3.png[/img]

Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$, respectively.

Suppose $ \vartriangle ABC$ is similar to $\vartriangle DEF$, with $ A$, $ B$, and $C$ corresponding to $D, E$, and $F$ respectively. If $\overline{AB} = \overline{EF}$, $\overline{BC} = \overline{FD}$, and $\overline{CA} = \overline{DE} = 2$, determine the area of $ \vartriangle ABC$.

Find the smallest real $\alpha$, such that for any convex polygon $P$ with area $1$, there exist a point $M$ in the plane, such that the area of convex hull of $P\cup Q$ is at most $\alpha$, where $Q$ denotes the image of $P$ under central symmetry with respect to $M$.