Found problems: 43
2012 Swedish Mathematical Competition, 6
A circle is inscribed in an trapezoid. Show that the diagonals of the trapezoid intersect at a point on the diameter of the circle perpendicular to the two parallel sides.
2020 Yasinsky Geometry Olympiad, 5
It is known that a circle can be inscribed in the quadrilateral $ABCD$, in addition $\angle A = \angle C$. Prove that $AB = BC$, $CD = DA$.
(Olena Artemchuk)
1986 Tournament Of Towns, (126) 1
We are given trapezoid $ABCD$ and point $M$ on the intersection of its diagonals. The parallel sides are $AD$ and $BC$ and it is known that $AB$ is perpendicular to $AD$ and that the trapezoid can have an inscribed circle. If the radius of this inscribed circle is $R$ find the area of triangle $DCM$ .
2023 CUBRMC, 4
Let square $ABCD$ and circle $\Omega$ be on the same plane, and $AA'$, $BB'$, $CC'$, $DD'$ be tangents to $\Omega$. Let $WXY Z$ be a convex quadrilateral with side lengths $WX = AA'$, $XY = BB'$, $Y Z = CC'$, and $ZW = DD'$. If $WXY Z$ has an inscribed circle, prove that the diagonals $WY$ and $XZ$ are perpendicular to each other.
1964 German National Olympiad, 6
Which of the following four statements are true and which are false?
a) If a polygon inscribed in a circle is equilateral, then it is also equiangular.
b) If a polygon inscribed in a circle is equiangular, then it is also equilateral.
c) If a polygon circumscribed to a circle is equilateral, then it is also equiangular.
d) If a polygon circumscribed to a circle is equiangular, then it is also equilateral.
Ukrainian From Tasks to Tasks - geometry, 2010.13
You can inscribe a circle in the pentagon $ABCDE$. It is also known that $\angle ABC = \angle BAE = \angle CDE = 90^o$. Find the measure of the angle $ADB$.
2020 OMpD, 3
Let $ABCD$ be a quadrilateral and let $\Gamma$ be a circle of center $O$ that is internally tangent to its four sides. If $M$ is the midpoint of $AC$ and $N$ is the midpoint of $BD$, prove that $M,O, N$ are collinear.
1957 Poland - Second Round, 6
Prove that if a convex quadrilateral has the property that there exists a circle tangent to its sides (i.e. an inscribed circle), and also a circle tangent to the extensions of its sides (an excircle), then the diagonals of the quadrilateral are perpendicular to each other.
1999 Romania National Olympiad, 3
In the convex quadrilateral $ABCD$, the bisectors of angles $A$ and $C$ intersect in $I$. Prove that $ABCD$ is circumscriptible if and only if
$$S[AIB] + S[CID] =S[AID]+S[BIC]$$
( $S[XYZ]$ denotes the area of the triangle $XYZ$)
2008 China Northern MO, 1A
As shown in figure , $\odot O$ is the inscribed circle of trapezoid $ABCD$, and the tangent points are $E, F, G, H$, $AB \parallel CD$. The line passing through$ B$, parallel to $AD$ intersects extension of $DC$ at point $P$. The extension of $AO$ intersects $CP$ at point $Q$. If $AE=BE$ , prove that $\angle CBQ = \angle PBQ$.
[img]https://cdn.artofproblemsolving.com/attachments/d/2/7c3a04bb1c59bc6d448204fd78f553ea53cb9e.png[/img]
1989 Tournament Of Towns, (214) 2
It is known that a circle can be inscribed in a trapezium $ABCD$.
Prove that the two circles, constructed on its oblique sides as diameters, touch each other.
(D. Fomin, Leningrad)
2009 Belarus Team Selection Test, 3
Given trapezoid $ABCD$ ($AD\parallel BC$) with $AD \perp AB$ and $T=AC\cap BD$. A circle centered at point $O$ is inscribed in the trapezoid and touches the side $CD$ at point $Q$. Let $P$ be the intersection point (different from $Q$) of the side $CD$ and the circle passing through $T,Q$ and $O$. Prove that $TP \parallel AD$.
I. Voronovich
2022 IFYM, Sozopol, 3
Quadrilateral $ABCD$ is circumscribed around circle $k$. Gind the smallest possible value of
$$\frac{AB + BC + CD + DA}{AC + BD}$$, as well as all quadrilaterals with the above property where it is reached.
1976 Chisinau City MO, 132
Let $O$ be the center of a circle inscribed in a convex quadrilateral $ABCD$ and $|AB|= a$, $|CD|=$c.
Prove that $$\frac{a}{c}=\frac{AO\cdot BO}{CO\cdot DO}.$$
1998 Tournament Of Towns, 3
Nine numbers are arranged in a square table:
$a_1 \,\,\, a_2 \,\,\,a_3$
$b_1 \,\,\,b_2 \,\,\,b_3$
$c_1\,\,\, c_2 \,\,\,c_3$ .
It is known that the six numbers obtained by summing the rows and columns of the table are equal:
$a_1 + a_2 + a_3 = b_1 + b_2 + b_3 = c_1 + c_2 + c_3 = a_1 + b_1 + c_1 = a_2 + b_2 + c_2 = a_3 + b_3 + c_3$ .
Prove that the sum of products of numbers in the rows is equal to the sum of products of numbers in the columns:
$a_1 b_1 c_1 + a_2 b_2c_2 + a_3b_3c_3 = a_1a_2a_3 + b_1 b_2 b_3 + c_1 c_2c_3$ .
(V Proizvolov)
1964 All Russian Mathematical Olympiad, 050
The quadrangle $ABCD$ is circumscribed around the circle with the centre $O$. Prove that $$\angle AOB+ \angle COD=180^o. $$
VI Soros Olympiad 1999 - 2000 (Russia), 11.3
A convex quadrilateral $ABCD$ has an inscribed circle touching its sides $AB$, $BC$, $CD$, $DA$ at the points $M$,$N$,$P$,$K$, respectively. Let $O$ be the center of the inscribed circle, the area of the quadrilateral $MNPK$ is equal to $8$. Prove the inequality $$2S \le OA \cdot OC+ OB \cdot OD.$$
2010 Singapore Junior Math Olympiad, 1
Let the diagonals of the square $ABCD$ intersect at $S$ and let $P$ be the midpoint of $AB$. Let $M$ be the intersection of $AC$ and $PD$ and $N$ the intersection of $BD$ and $PC$. A circle is incribed in the quadrilateral $PMSN$. Prove that the radius of the circle is $MP- MS$.