Found problems: 27
Inside an equilateral triangle there is a point whose distances from the sides of the triangle are $3, 4$ and $5$. Find the area of the triangle.
Let $\mathbf T$ be a triangle with $[\mathbf T]=1.$ Show that there is a right triangle $\mathbf R$ such that $[\mathbf R]\le\sqrt3$ and $\mathbf T\subseteq\mathbf R.$ ($[-]$ denotes area of a triangle.)
In the plane are given $27$ points, no three of which are collinear. Four of this points are vertices of a unit square, while the others lie inside the square. Prove that there are three points in this set forming a triangle with area not exceeding $1/48$.
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]
Through an arbitrary point inside a triangle, lines parallel to the sides of the triangle are drawn, dividing the triangle into three triangles with areas $T_1,T_2,T_3$ and three parallelograms. If $T$ is the area of the original triangle, prove that
$$T=(\sqrt{T_1}+\sqrt{T_2}+\sqrt{T_3})^2$$
.
In triangle $ABC$ with $\overline{AB}=\overline{AC}=3.6$, a point $D$ is taken on $AB$ at a distance $1.2$ from $A$. Point $D$ is joined to $E$ in the prolongation of $AC$ so that triangle $AED$ is equal in area to $ABC$. Then $\overline{AE}$ is:
$ \textbf{(A)}\ 4.8 \qquad\textbf{(B)}\ 5.4\qquad\textbf{(C)}\ 7.2\qquad\textbf{(D)}\ 10.8\qquad\textbf{(E)}\ 12.6 $
Let $K, L, M$, and $N$ be the midpoints of $CD,DA,AB$ and $BC$ of a square $ABCD$ respectively. Find the are of the triangles $AKB, BLC, CMD$ and $DNA$ if the square $ABCD$ has area $1$.
Let $180^o < \theta_1 < \theta_2 <...< \theta_n = 360^o$. For $i = 1,2,..., n$, $P_i = (\cos \theta_i^o, \sin \theta_i^o)$ is a point on the circle $C$ with centre $(0,0)$ and radius $1$. Let $P$ be any point on the upper half of $C$. Find the coordinates of $P$ such that the sum of areas $[PP_1P_2] + [PP_2P_3] + ...+ [PP_{n-1}P_n]$ attains its maximum.
In an acute-angled triangle $ABC$, the altitudes $AA_1,BB_1,CC_1$ intersect at point $V$. If the triangles $AC_1V, BA_1V, CB_1V$ have the same area, does it follow that the triangle $ABC$ is equilateral?
A triangle $ABC$ is given. For every positive numbers $p,q,r$, let $A',B',C'$ be the points such that $\overrightarrow{BA'} = p\overrightarrow{AB}, \overrightarrow{CB'}=q\overrightarrow{BC} $, and $\overrightarrow{AC'}=r\overrightarrow{CA}$. Define $f(p,q,r)$ as the ratio of the area of $\vartriangle A'B'C'$ to that of $\vartriangle ABC$. Prove that for all positive numbers $x,y,z$ and every positive integer $n$, $\sum_{k=0}^{n-1}f(x+k,y+k,z+k) = n^3f\left(\frac{x}{n},\frac{y}{n},\frac{z}{n}\right)$.
The diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ intersect at $O$. If $S_1$ and $S_2$ are the areas of triangles $AOB$ and $COD$ and S that of $ABCD$, show that $\sqrt{S_1} + \sqrt{S_2} \le \sqrt{S}$. Prove that equality holds if and only if $AB$ and $CD$ are parallel.
In a triangle $ABC$, the bisector of the largest angle $\angle A$ meets $BC$ at point $D$. Let $E$ and $F$ be the feet of perpendiculars from $D$ to $AC$ and $AB$, respectively. Let $R$ denote the ratio between the areas of triangles $DEB$ and $DFC$.
(a) Prove that, for every real number $r > 0$, one can construct a triangle ABC for which $R$ is equal to $r$.
(b) Prove that if $R$ is irrational, then at least one side length of $\vartriangle ABC$ is irrational.
(c) Give an example of a triangle $ABC$ with exactly two sides of irrational length, but with rational $R$.
$C$, $C'$ are concentric circles with radii $R$, $R'$. A rectangle has two adjacent vertices on $C$ and the other two vertices on $C'$. Find its sides if its area is as large as possible.
In an acute-angled triangle $ABC$ the altitudes $AU,BV,CW$ intersect at $H$.
Points $X,Y,Z$, different from $H$, are taken on segments $AU,BV$, and $CW$, respectively.
(a) Prove that if $X,Y,Z$ and $H$ lie on a circle, then the sum of the areas of triangles $ABZ, AYC, XBC$ equals the area of $ABC$.
(b) Prove the converse of (a).
Let $A_1A_2A_3$ be a triangle and $M$ an interior point. The straight lines $MA_1, MA_2, MA_3$ intersect the opposite sides at the points $B_1, B_2, B_3$ respectively (see Fig.). Show that if the areas of triangles $A_2B_1M, A_3B_2M$ and $A_1B_3M$ are equal, then $M$ coincides with the centroid of triangle $A_1A_2A_3$.
[img]https://cdn.artofproblemsolving.com/attachments/1/7/b29bdbb1f2b103be1f3cb2650b3bfff352024a.png[/img]
In a plane $5$ points are given such that all triangles having vertices at these points are of area not greater than $1$. Show that there exists a trapezoid which contains all point in the interior (or on the sides) and having the area not exceeding $3$.
An isosceles triangle $ABC$ with $AB = AC$ and $\angle A = 30^o$ is inscribed in a circle with center $O$. Point $D$ lies on the shorter arc $AC$ so that $\angle DOC = 30^o$, and point $G$ lies on the shorter arc $AB$ so that $DG = AC$ and $AG < BG$. The line $BG$ intersects $AC$ and $AB$ at $E$ and $F$, respectively.
(a) Prove that triangle $AFG$ is equilateral.
(b) Find the ratio between the areas of triangles $AFE$ and $ABC$.
In a convex hexagon $ABCDEF, AB$ is parallel to $DE, BC$ is parallel to $EF$ and $CD$ is parallel to $FA$. Prove that the triangles $ACE$ and $BDF$ have the same area.
The vertices of a triangle are lattice points. There are no lattice points on the sides (apart from the vertices) and $n$ lattice points inside the triangle. Show that its area is $n + \frac12$. Find the formula for the general case where there are also $m$ lattice points on the sides (apart from the vertices).
Let $ABCD$ be a convex quadrilateral and $A',B'C',D'$ be the midpoints of $AB,BC,CD,DA$, respectively. Let $a,b,c,d$ denote the areas of quadrilaterals into which lines $A'C'$ and $B'D'$ divide the quadrilateral $ABCD$ (where a corresponds to vertex $A$ etc.). Prove that $a+c = b+d$.
In a tetrahedron $VABC$, let $I$ be the incenter and $A',B',C'$ be arbitrary points on the edges $AV,BV,CV$, and let $S_a,S_b,S_c,S_v$ be the areas of triangles $VBC,VAC,VAB,ABC$, respectively. Show that points $A',B',C',I$ are coplanar if and only if $\frac{AA'}{A'V}S_a +\frac{BB'}{B'V}S_b +\frac{CC'}{C'V}S_c = S_v$
In the plane are given $2000$ congruent triangles of area $1$, which are all images of one triangle under translations. Each of these triangles contains the centroid of every other triangle. Prove that the union of these triangles has area less than $22/9$.
The diagonals $AC$ and $BD$ in the convex quadrilateral $ABCD$ intersect in $S$. Let $F_1$ and $F_2$ be the areas of $\vartriangle ABS$ and $\vartriangle CSD$. and let $F$ be the area of the quadrilateral $ABCD$. Show that $\sqrt{ F_1 }+\sqrt{ F_2}\le \sqrt{ F}$
If $P$ is the area of a triangle $ABC$ with sides $a,b,c$, prove that $\frac{ab+bc+ca}{4P} \ge \sqrt3$
Consider a unit square $ABCD$ and a (variable) equilateral triangle $XYZ$ such that $X, Z$ lie on rays $AB, DC,$ respectively, and $Y$ lies on segment $AD$. Compute the area of triangle $XYZ$ in terms of $x=AX$ and determine its maximum and minimum.