Found problems: 698
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 edges of the square in the figure have length $1$.
Find the area of the marked region in terms of $a$, where $0 \le a \le 1$.
[img]https://cdn.artofproblemsolving.com/attachments/2/2/f2b6ca973f66c50e39124913b3acb56feff8bb.png[/img]
Let $ABCD$ be a cyclic quadrilateral, $O$ be its circumcenter, $P$ be a common points of its diagonals, and $M , N$ be the midpoints of $AB$ and $CD$ respectively. A circle $OPM$ meets for the second time segments $AP$ and $BP$ at points $A_1$ and $B_1$ respectively and a circle $OPN$ meets for the second time segments $CP$ and $DP$ at points $C_1$ and $D_1$ respectively. Prove that the areas of quadrilaterals $AA_1B_1B$ and $CC_1D_1D$ are equal.
Let $P$ the intersection point of the diagonals of a convex quadrilateral $ABCD$. It is known that the area of triangles $ABC$, $BCD$ and $DAP$ is equal to $8 cm^2$, $9 cm^2$ and $10 cm^2$. Find the area of the quadrilateral $ABCD$.
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]
In a parallelogram $ABCD$ of surface area $60$ cm$^2$ , a line is drawn by $D$ that intersects $BC$ at $P$ and the extension of $AB$ at $Q$. If the area of the quadrilateral $ABPD$ is $46$ cm$^2$ , find the area of triangle $CPQ$.
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.
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.
From the triangle $ABC$, are gicen only the incenter $I$, the touchpoint $K$ of the inscribed circle with the side $AB$, as well as the center $I_a$ of the exscribed circle, that touches the side $BC$ . Construct a triangle equal in size to triangle $ABC$.
(Gryhoriy Filippovskyi)
Let $ABCDEF$ be a convex hexagon such that $BCEF$ is a parallelogram and $ABF$ an equilateral triangle. Given that $BC = 1, AD = 3, CD+DE = 2$, compute the area of $ABCDEF$
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:
$$
\operatorname{area}(DEF) = \frac{{PH}^2}{4R^2} \cdot \operatorname{area}(ABC).
$$
Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length $3$ and
the remaining four sides of length $2$. Give the answer in the form $r+s\sqrt{t}$ with $r,s, t$ positive integers.
Let $ABCD$ be a rectangle in which $AB + BC + CD = 20$ and $AE = 9$ where $E$ is the midpoint of the side $BC$. Find the area of the rectangle.
Let $d$ be a positive number. On the parabola, whose equation has the coefficient $1$ at the quadratic term, points $A, B$ and $C$ are chosen in such a way that the difference of the $x$-coordinates of points $A$ and $B$ is $d$ and the difference of the $x$-coordinates of points $B$ and $C$ is also $d$. Find the area of the triangle $ABC$.
Two points are located $10$ units apart, and a circle is drawn with radius $ r$ centered at one of the points. A tangent line to the circle is drawn from the other point. What value of $ r$ maximizes the area of the triangle formed by the two points and the point of tangency?
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 $A_E$ ?
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 quadrilateral $ABCD$ the sides $AB$ and $CD$ are parallel. Let $M$ be the midpoint of diagonal $AC$. Suppose that triangles $ABM$ and $ACD$ have equal area. Prove that $DM // BC$.
On the side $AB$ of the parallelogram $ABCD$ we take the points $X$ and $Y$ such that the points $A$, $X$, $Y$ and $B$ appear in this order. The lines $DX$ and $CY$ intersect at the point $Z$. Suppose that the area of the triangle $\triangle XYZ$ is equal to the sum of the areas of the triangles $\triangle AXD$ and $\triangle CYB$. Prove that the area of the quadrilateral $XYCD$ is equal to $3$ times the area of the triangle $\triangle XYZ$.
In the figure, there is a circle of radius $1$ such that the segment $AG$ is diameter and that line $AF$ is perpendicular to line $DC$. There are also two squares $ABDC$ and $DEGF$, where $B$ and $E$ are points on the circle, and the points $A$, $D$ and $E$ are collinear. What is the area of square $DEGF$?
[img]https://cdn.artofproblemsolving.com/attachments/1/e/794da3bc38096ef5d5daaa01d9c0f8c41a6f84.png[/img]
Consider the pentagon below. Find its area.
[img]https://cdn.artofproblemsolving.com/attachments/7/b/02ad3852b72682513cf62a170ed4aa45c23785.png[/img]
Inside the convex quadrilateral $ABCD$ lies the point $'M$. Reflect it symmetrically with respect to the midpoints of the sides of the quadrilateral and connect the obtained points so that they form a convex quadrilateral. Prove that the area of this quadrilateral does not depend on the choice of the point $M$.
Inside a circle $k$ of radius $R$ some round spots are made. The area of each spot is $1$. Every radius of circle $k$, as well as every circle concentric with $k$, meets in no more than one spot. Prove that the total area of all the spots is less than
$$\pi\sqrt R+\frac12R\sqrt R.$$
Given triangle $A_0B_0C_0$. On the segment $A_0B_0$ points $A_1$, $A_2$, $...$, $A_n$, and on the segment $B_0C_0$ - points $C_1$, $C_2$, $...$, $Cn$ so that all segments $A_iC_{i+1}$ ($i = 0$, $1$, $...$,$n-1$) are parallel to each other and all segments $ C_iA_{i+1}$ ($i = 0$, $1$, $...$,$n-1$) are too. Segments $C_0A_1$, $A_1C_2$, $A_2C_1$ and $C_1A_0$ bound a certain parallelogram, segments $C_1A_2$, $A_2C_3$, $A_3C_2$ and $C_2A_1$ too, etc. Prove that the sum of the areas of all $n -1$ resulting parallelograms less than half the area of triangle $A_0B_0C_0$.
Let $K$ be a convex polygon in the plane and suppose that $K$ is positioned in the coordinate system in such a way that
\[\text{area } (K \cap Q_i) =\frac 14 \text{area } K \ (i = 1, 2, 3, 4, ),\]
where the $Q_i$ denote the quadrants of the plane. Prove that if $K$ contains no nonzero lattice point, then the area of $K$ is less than $4.$