Inside a circle with radius $6$ lie four smaller circles with centres $A,B,C$ and $D$. The circles touch each other as shown. The point where the circles with centres $A$ and $C$ touch each other is the centre of the big circle. Calculate the area of quadrilateral $ABCD$. [img]https://1.bp.blogspot.com/-FFsiOOdcjao/XzT_oJYuQAI/AAAAAAAAMVk/PpyUNpDBeEIESMsiElbexKOFMoCXRVaZwCLcBGAsYHQ/s0/2012%2BMohr%2Bp1.png[/img]

In triangle $ABC$, $\angle BAC= 60^o$. Point $P$ is taken inside the triangle so that angles $\angle APB=\angle BPC= \angle CP A=120^o$. It is known that $AP = a$. Find the area of triangle $BPC$.

In a triangle $ABC$, let $D$ and $E$ be points of the sides $ BC$ and $AC$ respectively. Segments $AD$ and $BE$ intersect at $O$. Suppose that the line connecting midpoints of the triangle and parallel to $AB$, bisects the segment $DE$. Prove that the triangle $ABO$ and the quadrilateral $ODCE$ have equal areas.

Two circles intersect at two points $A$ and $B$. The radii of these circles are equal to $R$ and $r$, respectively; the angle between the radii going to the points of intersection is equal to $a$. A chord $KM$ of length $b$ is taken in a circle of radius $r$. Straight lines $KA$, $KB$, $MA$ and $MB$ intersect the other circle for second time at four points. Find the area of the quadrilateral with vertices at these points.

From vertex $A$ in square $ABCD$ (of side length $1$ ) two lines are drawn , one intersecting side $BC$ and the other intersecting side $CD$. The angle between these lines is $\theta$. From vertices $B$ and $D$ we construct perpendiculars to each of these lines . Find the area of the quadrilateral whose vertices are the four feet of these perpendiculars.

Let $ABCD$ be a rhombus of sides $AB = BC = CD= DA = 13$. On the side $AB$ construct the rhombus $BAFE$ outside $ABCD$ and such that the side $AF$ is parallel to the diagonal $BD$ of $ABCD$. If the area of $BAFE$ is equal to $65$, calculate the area of $ABCD$.

Let $a, b, c, d$ be the lengths of consecutive sides of a quadrilateral, and $S$ its area. Prove that $S \le \frac{ (a + b)(c + d)}{4}$

Let $ M,N,P,Q $ be points on the sides $ AB,BC,CD,DA $ (respectively) of a convex quadrilateral $ ABCD $ so that: $$ \frac{MA}{MB} =\frac{NB}{NC} =\frac{PD}{PC} =\frac{QA}{QD}\neq 1 $$ Show that the area of $ MNPQ $ is half the area of $ ABCD $ if and only if $ ABD $ and $ BCD $ have equal areas. [i]Petre Asaftei[/i]

Three circles are pairwise tangent, with none of them lying inside another. The centres of the circles are the corners of a triangle with circumference $1$. What is the smallest possible value for the sum of the areas of the circles?

A pentagon (not necessarily convex) has all sides of length $1$ and its product of cosine of any four angles equal to zero. Find all possible values of the area of such a pentagon.

A triangle $ABC$ with area $1$ is given. Anja and Bernd are playing the following game: Anja chooses a point $X$ on side $BC$. Then Bernd chooses a point $Y$ on side $CA$ und at last Anja chooses a point $Z$ on side $AB$. Also, $X,Y$ and $Z$ cannot be a vertex of triangle $ABC$. Anja wants to maximize the area of triangle $XYZ$ and Bernd wants to minimize that area. What is the area of triangle $XYZ$ at the end of the game, if both play optimally?

$ A',B',C' $ are the feet of the heights of an acute-angled triangle $ ABC. $ Calculate $$ \frac{\text{area} (ABC)}{\text{area}\left( A'B'C'\right)} , $$ knowing that $ ABC $ and $ A'B'C' $ have the same center of mass. [i]Carmen[/i] and [i]Viorel Botea[/i]

Given big enough sheet of cross-lined paper with the side of the squares equal to $1$. We are allowed to cut it along the lines only. Prove that for every $m>12$ we can cut out a rectangle of the greater than $m$ area such, that it is impossible to cut out a rectangle of $m$ area from it.

The quadrilateral $ABCD$ is a square of sidelength $1$, and the points $E, F, G, H$ are the midpoints of the sides. Determine the area of quadrilateral $PQRS$. [img]https://1.bp.blogspot.com/--fMGH2lX6Go/XzcDqhgGKfI/AAAAAAAAMXo/x4NATcMDJ2MeUe-O0xBGKZ_B4l_QzROjACLcBGAsYHQ/s0/2000%2BMohr%2Bp1.png[/img]

In $\vartriangle ABC$, the points $D, E$ and $F$ lie on $AB, BC$ and $CA$ respectively. The line segments $AE, BF$ and $CD$ meet at the point $G$. Suppose that the area of each of $\vartriangle BGD, \vartriangle ECG$ and $\vartriangle GFA$ is $1$ cm$^2$. Prove that the area of each of $\vartriangle BEG, \vartriangle GCF$ and $\vartriangle ADG$ is also $1$ cm$^2$. [img]https://cdn.artofproblemsolving.com/attachments/e/7/ec090135bd2e47a9681d767bb984797d87218c.png[/img]

Angle bisector from vertex $A$ of acute triangle $ABC$ intersects side $BC$ in point $D$, and circumcircle of $ABC$ in point $E$ (different from $A$). Let $F$ and $G$ be foots of perpendiculars from point $D$ to sides $AB$ and $AC$. Prove that area of quadrilateral $AEFG$ is equal to the area of triangle $ABC$

The diagonals of some convex quadrilateral are mutually perpendicular and divide the quadrangle into $4$ triangles, the areas of which are expressed by prime numbers. Prove that a circle can be inscribed in this quadrilateral.

Let $ABC$ be a triangle with circumradius $R$, and let $\ell_A, \ell_B, \ell_C$ be the altitudes through $A, B, C$ respectively. The altitudes meet at $H$. Let $P$ be an arbitrary point in the same plane as $ABC$. The feet of the perpendicular lines through $P$ onto $\ell_A, \ell_B, \ell_C$ are $D, E, F$ respectively. Prove that the areas of $DEF$ and $ABC$ satisfy the following equation: $$ \operatorname{area}(DEF) = \frac{{PH}^2}{4R^2} \cdot \operatorname{area}(ABC). $$

Two circles in the plane, both of radius $R$, intersect at a right angle. Compute the area of the intersection of the interiors of the two circles.

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]

We are given a right triangle $ABC$ and the median $BD$ drawn from the vertex $B$ of the right angle. Let the circle inscribed in $\vartriangle ABD$ be tangent to side $AD$ at $K$. Find the angles of $\vartriangle ABC$ if $K$ divides $AD$ in halves.

Daniel finds a rectangular index card and measures its diagonal to be 8 centimeters. Daniel then cuts out equal squares of side 1 cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be $4\sqrt{2}$ centimeters, as shown below. What is the area of the original index card? [asy] unitsize(0.6 cm); pair A, B, C, D, E, F, G, H; real x, y; x = 9; y = 5; A = (0,y); B = (x - 1,y); C = (x - 1,y - 1); D = (x,y - 1); E = (x,0); F = (1,0); G = (1,1); H = (0,1); draw(A--B--C--D--E--F--G--H--cycle); draw(interp(C,G,0.03)--interp(C,G,0.97), dashed, Arrows(6)); draw(interp(A,E,0.03)--interp(A,E,0.97), dashed, Arrows(6)); label("$1$", (B + C)/2, W); label("$1$", (C + D)/2, S); label("$8$", interp(A,E,0.3), NE); label("$4 \sqrt{2}$", interp(G,C,0.2), SE); [/asy] $\textbf{(A) }14\qquad\textbf{(B) }10\sqrt{2}\qquad\textbf{(C) }16\qquad\textbf{(D) }12\sqrt{2}\qquad\textbf{(E) }18$

Fencing Ferdinand wants to fence three rectangular areas. there are fences in three types, with $4$ amount of fences of each type. You will notice that there is always at least as much area it manages to enclose a total of three by enclosing three square areas (i.e., each area fencing elements of the same size to enclose it) as if it were three different, rectangular would encircle an area (i.e., use two different elements for each of the three areas). Why is this is so? When does it not matter how he fences the rectangles, in terms of the sum of the areas?

Determine the least real number $a > 1$ such that for any point $P$ in the interior of a square $ABCD$, the ratio of the areas of some two triangle $PAB, PBC, PCD, PDA$ lies in the interval $[1/a, a]$.

Let $ABCA'B'C'$ a right triangular prism with the bases equilateral triangles. A plane $\alpha$ containing point $A$ intersects the rays $BB'$ and $CC'$ at points E and $F$, so that $S_ {ABE} + S_{ACF} = S_{AEF}$. Determine the measure of the angle formed by the plane $(AEF)$ with the plane $(BCC')$.