Found problems: 264
A rhombus with its incircle is given. At each vertex of the rhombus a circle is constructed that touches the incircle and two edges of the rhombus. These circles have radii $r_1,r_2$, while the incircle has radius $r$. Given that $r_1$ and $r_2$ are natural numbers and that $r_1r_2=r$, find $r_1,r_2,$ and $r$.
Given a rhombus $ABCD$. A point $M$ is chosen on its side $BC$. The lines, which pass through $M$ and are perpendicular to $BD$ and $AC$, meet line $AD$ in points $P$ and $Q$ respectively. Suppose that the lines $PB,QC,AM$ have a common point. Find all possible values of a ratio $\frac{BM}{MC}$.
[i]S. Berlov, F. Petrov, A. Akopyan[/i]
Two circles $ O, O'$ having same radius meet at two points, $ A,B (A \not = B) $. Point $ P,Q $ are each on circle $ O $ and $ O' $ $(P \not = A,B ~ Q\not = A,B )$. Select the point $ R $ such that $ PAQR $ is a parallelogram. Assume that $ B, R, P, Q $ is cyclic. Now prove that $ PQ = OO' $.
$ABCD$ 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 $CD$.
A rhombus is inscribed in triangle $ ABC$ in such a way that one of its vertices is $ A$ and two of its sides lie along $ AB$ and $ AC$. If $ \overline{AC} \equal{} 6$ inches, $ \overline{AB} \equal{} 12$ inches, and $ \overline{BC} \equal{} 8$ inches, the side of the rhombus, in inches, is:
$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 3 \frac {1}{2} \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{ \angle BAC, \angle ABC \}$.
Let $ABCD$ be a rhombus with $\angle BAD < 90^o$. The circle passing through $D$ with center $A$ intersects the line $CD$ a second time in point $E$. Let $S$ be the intersection of the lines $BE$ and $AC$. Prove that the points $A$, $S$, $D$ and $E$ lie on a circle.
[i](Karl Czakler)[/i]
Outside the square $ABCD$, the rhombus $BCMN$ is constructed with angle $BCM$ obtuse . Let $P$ be the intersection point of the lines $BM$ and $AN$ . Prove that $DM \perp CP$ and the triangle $DPM$ is right isosceles .
It is allowed to perform the following transformations in the plane with any integers $a$:
(1) Transform every point $(x, y)$ to the corresponding point $(x + ay, y)$,
(2) Transform every point $(x, y)$ to the corresponding point $(x, y + ax)$.
Does there exist a non-square rhombus whose all vertices have integer coordinates and which can be transformed to:
a) Vertices of a square,
b) Vertices of a rectangle with unequal side lengths?
Let $n>3$ be a positive integer. Equilateral triangle ABC is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$.
In the illustration, a regular hexagon and a regular octagon have been tiled with rhombuses.
In each case, the sides of the rhombuses are the same length as the sides of the regular polygon.
(a) Tile a regular decagon ($10$-gon) into rhombuses in this manner.
(b) Tile a regular dodecagon ($12$-gon) into rhombuses in this manner.
(c) How many rhombuses are in a tiling by rhombuses of a $2002$-gon?
Justify your answer.
[img]https://cdn.artofproblemsolving.com/attachments/8/a/8413e4e2712609eba07786e34ba2ce4aa72888.png[/img]
We have constructed a rhombus by attaching two equal equilateral triangles. By putting $n-1$ points on all 3 sides of each triangle we have divided the sides to $n$ equal segments. By drawing line segements between correspounding points on each side of the triangles we have divided the rhombus into $2n^2$ equal triangles.
We write the numbers $1,2,\dots,2n^2$ on these triangles in a way no number appears twice. On the common segment of each two triangles we write the positive difference of the numbers written on those triangles. Find the maximum sum of all numbers written on the segments.
(25 points)
[i]Proposed by Amirali Moinfar[/i]
Two equilateral triangles are contained in a square whose side length is $2\sqrt3$. The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus?
$ \textbf{(A)}\ \frac{3}{2}\qquad\textbf{(B)}\ \sqrt3\qquad\textbf{(C)}\ 2\sqrt2-1\qquad\textbf{(D)}\ 8\sqrt3-12\qquad\textbf{(E)}\ \frac{4\sqrt3}{3}$
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)
The figure below consists of five isosceles triangles and ten rhombi. The bases of the isosceles triangles are $12$, $13$, $14$, $15$, as shown below. The top rhombus, which is shaded, is actually a square. Find the area of this square.
[DIAGRAM NEEDED]
Prove that if all faces of a parallelepiped are equal parallelograms, they are rhombuses.
A square sheet of paper $ABCD$ is so folded that $B$ falls on the mid point of $M$ of $CD$. Prove that the crease will divide $BC$ in the ration $5 : 3$.
A quadrilateral $ ABCD $ is inscribed in a circle. The lines $AB$ and $CD$ intersect at point $E$, and the lines $AD$ and $BC$ intersect at point $F$. The bisector of the angle $ AEC $ intersects the side $ BC $ at the point $ M $ and the side $ AD $ at the point $ N $; and the bisector of the angle $ BFD $ intersects the side $ AB $ at the point $ P $ and the side $ CD $ at the point $ Q $. Prove that the quadrilateral $MPNQ$ is a rhombus.
The diagonals of a convex quadrilateral divide it into four triangles. The radii of the circles circumscribed around these triangles are equal. Can such a property have a quadrilateral other than:
a) parallelogram,
b) rhombus?
(Sharygin Igor)
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 $ABC$.
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.
Each of two different lines parallel to the the axis $Ox$ have exactly two common points on the graph of the function $f(x)=x^3+ax^2+bx+c$. Let $\ell_1$ and $\ell_2$ be two lines parallel to $Ox$ axis which meet the graph of $f$ in points $K_1, K_2$ and $K_3, K_4$, respectively. Prove that the quadrilateral formed by $K_1, K_2, K_3$ and $ K_4$ is a rhombus if and only if its area is equal to $6$ units.
A circle with radius $1$ is circumscribed by a rhombus. What is the minimum possible area of this rhombus?
$ABC$ is an isosceles triangle with $AB=AC$ and the angle in $A$ is less than $60^{\circ}$. Let $D$ be a point on $AC$ such that $\angle{DBC}=\angle{BAC}$. $E$ is the intersection between the perpendicular bisector of $BD$ and the line parallel to $BC$ passing through $A$. $F$ is a point on the line $AC$ such that $FA=2AC$ ($A$ is between $F$ and $C$).
Show that $EB$ and $AC$ are parallel and that the perpendicular from $F$ to $AB$, the perpendicular from $E$ to $AC$ and $BD$ are concurrent.
On the $BE$ side of a regular $ABE$ triangle, a $BCDE$ rhombus is built outside it. The segments $AC$ and $BD$ intersect at point $F$. Prove that $AF <BD$.