The point $E$ is the midpoint of the segment connecting the orthocentre of the scalene triangle $ABC$ and the point $A$. The incircle of triangle $ABC$ incircle is tangent to $AB$ and $AC$ at points $C'$ and $B'$ respectively. Prove that point $F$, the point symmetric to point $E$ with respect to line $B'C'$, lies on the line that passes through both the circumcentre and the incentre of triangle $ABC$.

A quadrilateral is circumscribed around a circle. Its diagonals intersect at the center of the circle. Prove that the quadrilateral is a rhombus.

Let $ABCD$ be a trapezoid with $AB // CD$, $2|AB| = |CD|$ and $BD \perp BC$. Let $M$ be the midpoint of $CD$ and let $E$ be the intersection $BC$ and $AD$. Let $O$ be the intersection of $AM$ and $BD$. Let $N$ be the intersection of $OE$ and $AB$. (a) Prove that $ABMD$ is a rhombus. (b) Prove that the line $DN$ passes through the midpoint of the line segment $BE$.

Let $ABCD$ be a rhombus, let $APQC$ be a parallelogram such that the point $B$ lies inside it and the side $AP$ is equal to the side of the rhombus. Prove that $B$ is the orthocenter of the triangle $DPQ$. Egor Bakaev

What is the radius of a circle inscribed in a rhombus with diagonals of length $ 10$ and $ 24$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 58/13 \qquad \textbf{(C)}\ 60/13 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

Let $ABCD$ be a convex quadrilateral. Diagonals $AC$ and $BD$ meet at point $P$. The inradii of triangles $ABP$, $BCP$, $CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus.

$(YUG 3)$ Let four points $A_i (i = 1, 2, 3, 4)$ in the plane determine four triangles. In each of these triangles we choose the smallest angle. The sum of these angles is denoted by $S.$ What is the exact placement of the points $A_i$ if $S = 180^{\circ}$?

On a map whose scale is $ 400$ miles to an inch and a half, a certain estate is represented by a rhombus having a $ 60^{\circ}$ angle. The diagonal opposite $ 60^{\circ}$ is $ \frac {3}{16}$ in. The area of the estate in square miles is: $ \textbf{(A)}\ \frac {2500}{\sqrt {3}} \qquad\textbf{(B)}\ \frac {1250}{\sqrt {3}} \qquad\textbf{(C)}\ 1250 \qquad\textbf{(D)}\ \frac {5625\sqrt {3}}{2} \qquad\textbf{(E)}\ 1250\sqrt {3}$

Let $ a,b$ be two complex numbers. Prove that roots of $ z^{4}\plus{}az^{2}\plus{}b$ form a rhombus with origin as center, if and only if $ \frac{a^{2}}{b}$ is a non-positive real number.

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 $ABCD$.

$ABC$ is an acute angle triangle such that $AB>AC$ and $\hat{BAC}=60^{\circ}$. Let's denote by $O$ the center of the circumscribed circle of the triangle and $H$ the intersection of altitudes of this triangle. Line $OH$ intersects $AB$ in point $P$ and $AC$ in point $Q$. Find the value of the ration $\frac{PO}{HQ}$.

A convex quadrangle $ ABCD$ is inscribed in a circle with center $ O$. The angles $ AOB, BOC, COD$ and $ DOA$, taken in some order, are of the same size as the angles of the quadrangle $ ABCD$. Prove that $ ABCD$ is a square

Let $E$ be a point inside the rhombus $ABCD$ such that $|AE|=|EB|$, $m(\widehat{EAB})=12^\circ$, and $m(\widehat{DAE})=72^\circ$. What is $m(\widehat{CDE})$ in degrees? $ \textbf{(A)}\ 64 \qquad\textbf{(B)}\ 66 \qquad\textbf{(C)}\ 68 \qquad\textbf{(D)}\ 70 \qquad\textbf{(E)}\ 72 $

Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.

Let $ABCD$ be a convex quadrilateral. Prove that the incircles of the triangles $ABC$, $BCD$, $CDA$ and $DAB$ have a point in common if, and only if, $ABCD$ is a rhombus.

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 $ABCD$.

$ABCD$ is a regular tetrahedron with side length $1$. Find the area of the cross section of $ABCD$ cut by the plane that passes through the midpoints of $AB$, $AC$, and $CD$.

$ABCD$ is a fixed rhombus. Segment $PQ$ is tangent to the inscribed circle of $ABCD$, where $P$ is on side $AB$, $Q$ is on side $AD$. Show that, when segment $PQ$ is moving, the area of $\Delta CPQ$ is a constant.

Let $P$ be a regular $2n$-sided polygon. A [b]rhombus-ulation[/b] of $P$ is dividing $P$ into rhombuses such that no two intersect and no vertex of any rhombus is on the edge of other rhombuses or $P$. (a) Prove that number of rhombuses is a function of $n$. Find the value of this function. Also find the number of vertices and edges of the rhombuses as a function of $n$. (b) Prove or disprove that there always exists an edge $e$ of $P$ such that by erasing all the segments parallel to $e$ the remaining rhombuses are connected. (c) Is it true that each two rhombus-ulations can turn into each other using the following algorithm multiple times? Algorithm: Take a hexagon -not necessarily regular- consisting of 3 rhombuses and re-rhombus-ulate the hexagon. (d) Let $f(n)$ be the number of ways to rhombus-ulate $P$. Prove that:\[\Pi_{k=1}^{n-1} ( \binom{k}{2} +1) \leq f(n) \leq \Pi_{k=1}^{n-1} k^{n-k} \]

Let $ABCDEF$ be a convex hexagon satisfying $AC=DF$, $CE=FB$ and $EA=BD$. Prove that the lines connecting the midpoints of opposite sides of the hexagon $ABCDEF$ intersect in one point.

Let $ABCD$ be a rhombus and $P$ be a point on its side $BC$. The circle passing through $A, B$, and $P$ intersects $BD$ once more at the point $Q$ and the circle passing through $C,P$ and $Q$ intersects $BD$ once more at the point $R$. Prove that $A, R$ and $P$ lie on the one straight line. (D. Fomin, Leningrad)

Let $P$ be a point outside a circle $\omega$. The tangents from $P$ to $\omega$ are drawn touching $\omega$ at points $A$ and $B$. Let $M$ and $N$ be the midpoints of $AP$ and $AB$, respectively. Line $MN$ is extended to cut $\omega$ at $C$ so that $N$ lies between $M$ and $C$. Line $PC$ intersects $\omega$ again at $D$, and lines $ND$ and $PB$ intersect at $O$. Prove that $MNOP$ is a rhombus.

The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\frac{1}{2}\left(\sqrt{p}-q\right),$ where $p$ and $q$ are positive integers. Find $p+q.$ [asy] size(250);real x=sqrt(3); int i; draw(origin--(14,0)--(14,2+2x)--(0,2+2x)--cycle); for(i=0; i<7; i=i+1) { draw(Circle((2*i+1,1), 1)^^Circle((2*i+1,1+2x), 1)); } for(i=0; i<6; i=i+1) { draw(Circle((2*i+2,1+x), 1)); }[/asy]

In the figure below, each side of the rhombus has length 5 centimeters. [asy] import graph; unitsize(2.0cm); real w = sqrt(3); draw((w, 0) -- (0, 1) -- (-w, 0) -- (0, -1) -- cycle); filldraw(Circle((-0.5, 0), 0.8 / sqrt(pi)), gray); label("$60^\circ$", (w - 0.1, 0), W); [/asy] The circle lies entirely within the rhombus. The area of the circle is $n$ square centimeters, where $n$ is a positive integer. Compute the number of possible values of $n$.

A circle cuts each side of a rhombus twice thus dividing each side into three segments. Let us go around the perimeter of the rhombus clockwise beginning at a vertex and paint these segments successively in red, white and blue. Prove that the sum of lengths of the blue segments equals that of the red ones. (V Proizvolov)