Given a rectangle $ABCD$ such that $AB = b > 2a = BC$, let $E$ be the midpoint of $AD$. On a line parallel to $AB$ through point $E$, a point $G$ is chosen such that the area of $GCE$ is $$(GCE)= \frac12 \left(\frac{a^3}{b}+ab\right)$$ Point $H$ is the foot of the perpendicular from $E$ to $GD$ and a point $I$ is taken on the diagonal $AC$ such that the triangles $ACE$ and $AEI$ are similar. The lines $BH$ and $IE$ intersect at $K$ and the lines $CA$ and $EH$ intersect at $J$. Prove that $KJ \perp AB$.

We draw two lines $(\ell_1) , (\ell_2)$ through the orthocenter $H$ of the triangle $ABC$ such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle.

Let $ABC$ be a triangle, and $M$ be the midpoint of $BC$, Let $N$ be the point on the segment $AM$ with $AN = 3NM$, and $P$ be the intersection point of the lines $BN$ and $AC$. What is the area in cm$^2$ of the triangle $ANP$ if the area of the triangle $ABC$ is $40$ cm$^2$?

For a triangle $ ABC,$ let $ k$ be its circumcircle with radius $ r.$ The bisectors of the inner angles $ A, B,$ and $ C$ of the triangle intersect respectively the circle $ k$ again at points $ A', B',$ and $ C'.$ Prove the inequality \[ 16Q^3 \geq 27 r^4 P,\] where $ Q$ and $ P$ are the areas of the triangles $ A'B'C'$ and $ABC$ respectively.

The sides of $\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron?

Let be given a triangle $ABC$ and points $D,M,N$ belong to $BC,AB,AC$, respectively. Suppose that $MD$ is parallel to $AC$ and $ND$ is parallel to $AB$. If $S_{\vartriangle BMD} = 9$ cm $^2, S_{\vartriangle DNC} = 25$ cm$^2$, compute $S_{\vartriangle AMN}$?

Given a pentagon of area $1993$ and $995$ points inside the pentagon, let $S$ be the set containing the vertices of the pentagon and the $995$ points. Show that we can find three points of $S$ which form a triangle of area $\le 1$.

$ABCD$ is a square of center $O$. On the sides $DC$ and $AD$ the equilateral triangles DAF and DCE have been constructed. Decide if the area of the $EDF$ triangle is greater, less or equal to the area of the $DOC$ triangle. [img]https://4.bp.blogspot.com/-o0lhdRfRxl0/XNYtJgpJMmI/AAAAAAAAKKg/lmj7KofAJosBZBJcLNH0JKjW3o17CEMkACK4BGAYYCw/s1600/may4_2.gif[/img]

Let $ABC$ be a triangle with $AB = 3$, $BC = 5$, $AC = 7$, and let $ P$ be a point in its interior. If $G_A$, $G_B$, $G_C$ are the centroids of $\vartriangle PBC$, $\vartriangle PAC$, $\vartriangle PAB$, respectively, find the maximum possible area of $\vartriangle G_AG_BG_C$.

Let $ABC$ be a triangle. Points $D$, $E$, and $F$ are respectively on the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ of $\triangle ABC$. Suppose that \[ \frac{AE}{AC} = \frac{CD}{CB} = \frac{BF}{BA} = x \] for some $x$ with $\frac{1}{2} < x < 1$. Segments $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ cut the triangle into 7 nonoverlapping regions: 4 triangles and 3 quadrilaterals. The total area of the 4 triangles equals the total area of the 3 quadrilaterals. Compute the value of $x$.

Let $ ABC$ be an acute-angled triangle. Let $ L$ be any line in the plane of the triangle $ ABC$. Denote by $ u$, $ v$, $ w$ the lengths of the perpendiculars to $ L$ from $ A$, $ B$, $ C$ respectively. Prove the inequality $ u^2\cdot\tan A \plus{} v^2\cdot\tan B \plus{} w^2\cdot\tan C\geq 2\cdot S$, where $ S$ is the area of the triangle $ ABC$. Determine the lines $ L$ for which equality holds.

In the right triangle $ABC$ with shorter side $AC$ the hypotenuse $AB$ has length $12$. Denote $T$ its centroid and $D$ the feet of altitude from the vertex $C$. Determine the size of its inner angle at the vertex $B$ for which the triangle $DTC$ has the greatest possible area.

A square and an equilateral triangle have the same perimeter. Let $ A$ be the area of the circle circumscribed about the square and $ B$ be the area of the circle circumscribed about the triangle. Find $ A/B$. $ \textbf{(A)}\ \frac{9}{16} \qquad \textbf{(B)}\ \frac{3}{4} \qquad \textbf{(C)}\ \frac{27}{32} \qquad \textbf{(D)}\ \frac{3\sqrt{6}}{8} \qquad \textbf{(E)}\ 1$

Two right triangles are given, of which the incircle of the first triangle is the circumcircle of the second triangle. Let the areas of the triangles be $S$ and $S'$ respectively. Prove that $\frac{S}{S'} \ge 3 +2\sqrt2$

Circle $k$ passes through $A$ and intersects the sides of $\Delta ABC$ in $P,Q$, and $L$. Prove that: $\frac{S_{PQL}}{S_{ABC}}\leq \frac{1}{4} (\frac{PL}{AQ})^2$.

Let $a,b,c$ be positive real numbers such that $abc(a+b+c)=3.$ Prove that we have \[(a+b)(b+c)(c+a)\geq 8.\] Also determine the case of equality.

Let $T_1$ be a triangle having $a, b, c$ as lengths of its sides and let $T_2$ be another triangle having $u, v,w$ as lengths of its sides. If $P,Q$ are the areas of the two triangles, prove that \[16PQ \leq a^2(-u^2 + v^2 + w^2) + b^2(u^2 - v^2 + w^2) + c^2(u^2 + v^2 - w^2).\] When does equality hold?

In this exercise we only consider convex quadrilaterals. (a) For such a quadrilateral $ABCD$, determine the set of points $P$ contained within that quadrilateral for which $PA$ and $PC$ divide the quadrilateral into two pieces of equal areas. (b) Prove that there is a point $P$ inside such a quadrilateral, such that the triangles $PAB$ and $PCD$ have equal areas, as well as the triangles $PBC$ and $PAD$. (c) Find out which quadrilaterals $ABCD$ contains a point $P$, so that the triangles $PAB$, $PBC$, $PCD$ and $PDA$ have equal areas.

Let $a, b, c$ be the side lengths and $P$ be area of a triangle, respectively. Prove that \[(a^2+b^2+c^2-4\sqrt 3 P) (a^2+b^2+c^2) \geq 2 \left(a^2(b - c)^2 + b^2(c - a)^2 + c^2(a - b)^2\right).\]

Let $ABCD$ be a trapezium (trapezoid) with $AD$ parallel to $BC$ and $J$ be the intersection of the diagonals $AC$ and $BD$. Point $P$ a chosen on the side $BC$ such that the distance from $C$ to the line $AP$ is equal to the distance from $B$ to the line $DP$. [i]The following three questions 1, 2 and 3 are independent, so that a condition in one question does not apply in another question.[/i] 1.Suppose that $Area( \vartriangle AJB) =6$ and that $Area(\vartriangle BJC) = 9$. Determine $Area(\vartriangle APD)$. 2. Find all points $Q$ on the plane of the trapezium such that $Area(\vartriangle AQB) = Area(\vartriangle DQC)$. 3. Prove that $PJ$ is the angle bisector of $\angle APD$.

Let $k_1, k_2$ and $k_3$ be three circles with centers $O_1, O_2$ and $O_3$ respectively, such that no center is inside of the other two circles. Circles $k_1$ and $k_2$ intersect at $A$ and $P$, circles $k_1$ and $k_3$ intersect and $C$ and $P$, circles $k_2$ and $k_3$ intersect at $B$ and $P$. Let $X$ be a point on $k_1$ such that the line $XA$ intersects $k_2$ at $Y$ and the line $XC$ intersects $k_3$ at $Z$, such that $Y$ is nor inside $k_1$ nor inside $k_3$ and $Z$ is nor inside $k_1$ nor inside $k_2$. a) Prove that $\triangle XYZ$ is simular to $\triangle O_1O_2O_3$ b) Prove that the $P_{\triangle XYZ} \le 4P_{\triangle O_1O_2O_3}$. Is it possible to reach equation?$ *Note: $P$ denotes the area of a triangle*

In a convex pentagon $ABCDE$, the area of the triangles $ABC, ABD, ACD$ and $ADE$ are equal and have the value $F$. What is the area of the triangle $BCE$ ?

Given a regular triangle $ABC$, whose area is $1$, and the point $P$ on its circumscribed circle. Lines $AP, BP, CP$ intersect, respectively, lines $BC, CA, AB$ at points $A', B', C'$. Find the area of the triangle $A'B'C'$.

The $ABC$ triangle is given, point $I_a$ is the center of an exscribed circle touching the side $BC$ , the point $M$ is the midpoint of the side $BC$, the point $W$ is the intersection point of the bisector of the angle $A$ of the triangle $ABC$ with the circumscribed circle around him. Prove that the area of the triangle $I_aBC$ is calculated by the formula $S_{ (I_aBC)} = MW \cdot p$, where $p$ is the semiperimeter of the triangle $ABC$. (Mykola Moroz)

Given are two points $B,C$. Consider point $A$ not lying on the line $BC$ and draw the circles $C_1(K_1,R_1)$ (with center $K_1$ and radius $R_1$) and $C_2(K_2,R_2)$ with chord $AB, AC$ respectively such that their centers lie on the interior of the triangle $ABC$ and also $R_1 \cdot AC= R_2 \cdot AB$. Let $T$ be the intersection point of the two circles, different from $A$, and M be a random pointof line $AT$, prove that $TC \cdot S_{(MBT)}=TB \cdot S_{(MCT)}$