Consider quadrilateral $ABCD$ with $|DC| > |AD|$. Let $P$ be a point on $DC$ such that $PC = AD$ and let $Q$ be the midpoint of $DP$. Let $L_1$ be the line perpendicular on $DC$ passing through $Q$ and let $L_2$ be the bisector of the angle $ \angle ABC$. Let us call $X = L_1 \cap L_2$. Show that if quadrilateral is cyclic then $X$ lies on the circumcircle of $ABCD.$ [img]https://cdn.artofproblemsolving.com/attachments/f/6/3ebfce8a7fd2a0ece9f09065608141006893d2.png[/img]

Let $ABCD$ be a convex cyclic quadrilateral with diagonals $AC$ and $BD$. Each of the four vertixes are reflected across the diagonal on which the do not lie. (a) Investigate when the four points thus obtained lie on a straight line and give as simple an equivalent condition as possible to the cyclic quadrilateral $ABCD$ for it. (b) Show that in all other cases the four points thus obtained lie on one circle. (Theresia Eisenkölbl)

Let O be the circumcenter of triangle ABC. Let H be the orthocenter of triangle ABC. the perpendicular bisector of AB meet AC,BC at D,E. the circumcircle of triangle DEH meet AC,BC,OH again at F,G,L. CH meet FG at T,and ABCT is concyclic. Prove that LHBC is concyclic. graph: https://cdn.luogu.com.cn/upload/image_hosting/w6z6mvm4.png

Let $\triangle ABC$ be a triangle and let $X$ be a point in the interior of the triangle. The second intersection points of the lines $XA,XB$ and $XC$ with the circumcircle of $\triangle ABC$ are $P,Q$ and $R$. Let $U$ be a point on the ray $XP$ (these are the points on the line $XP$ such that $P$ and $U$ lie on the same side of $X$). The line through $U$ parallel to $AB$ intersects $BQ$ in $V$ . The line through $U$ parallel to $AC$ intersects $CR$ in $W$. Prove that $Q, R, V$ , and $W$ lie on a circle.

On the $ AB $ side of the triangle $ ABC $, points $ X $ and $ Y $ are chosen, on the side of $ AC $ is a point of $ Z $, and on the side of $ BC $ is a point of $ T $. Wherein $ XZ \parallel BC $, $ YT \parallel AC $. Line $ TZ $ intersects the circumscribed circle of triangle $ ABC $ at points $ D $ and $ E $. Prove that points $ X $, $ Y $, $ D $ and $ E $ lie on the same circle.

Let ABC be an acute isosceles triangle ($AB=BC$) inscribed in a circle with center $O$ . The line through the midpoint of the chord $AB$ and point $O$ intersects the line $AC$ at $L$ and the circle at the point $P$. Let the bisector of angle $BAC$ intersects the circle at point $K$. Lines $AB$ and $PK$ intersect at point $D$. Prove that the points $L,B,D$ and $P$ lie on the same circle.

Two circles $ w_1 $ and $ w_2 $ intersect at two points $ P $ and $ Q $. The common tangent to $ w_1 $ and $ w_2 $, which is closer to the point $ P $ than to $ Q $, touches these circles at $ A $ and $ B $, respectively. The tangent to $ w_1 $ at the point $ P $ intersects $ w_2 $ at the point $ E $ (different from $ P $), and the tangent to $ w_2 $ at the point $ P $ intersects $ w_1 $ at $ F $ (different from $ P $). Let $ H $ and $ K $ be points on the rays $ AF $ and $ BE $, respectively, such that $ AH = AP $ and $ BK = BP $. Prove that the points $ A $, $ H $, $ Q $, $ K $ and $ B $ lie on the same circle.

The triangle $ABC$ with $AB> AC$ is inscribed in a circle, the angle bisector of $\angle BAC$ intersects the side $BC$ of the triangle at the point $K$, and the circumscribed circle at the point $M$. The midline of $\Delta ABC$, which is parallel to the side $AB$, intersects $AM$ at the point $O$, the line $CO$ intersects the line $AB$ at the point $N$. Prove that a circle can be circumscribed around the quadrilateral $BNKM$. (Nagel Igor)

Let $ABC$ be a triangle in which $AB < AC, D$ is the foot of the altitude from $A, H$ is the orthocenter, $O$ is the circumcenter, $M$ is the midpoint of the side $BC, A'$ is the reflection of $A$ across $O$, and $S$ is the intersection of the tangents at $B$ and $C$ to the circumcircle. The tangent at $A'$ to the circumcircle intersects $SC$ and $SB$ at $X$ and $Y$ , respectively. If $M,S,X,Y$ are concyclic, prove that lines $OD$ and $SA'$ are parallel.

Let $AA_1, BB_1, CC_1$ be the heights of triangle $ABC$ and $H$ be its orthocenter. Liune $\ell$ parallel to $AC$, intersects straight lines $AA_1$ and $CC_1$ at points $A_2$ and $C_2$, respectively. Suppose that point $B_1$ lies outside the circumscribed circle of triangle $A_2 HC_2$. Let $B_1P$ and $B_1T$ be tangent to of this circle. Prove that points $A_1, C_1, P$, and $T$ are cyclic.

The projections of the point $X$ onto the sides of the $ABCD$ quadrangle lie on the same circle. $Y$ is a point symmetric to $X$ with respect to the center of this circle. Prove that the projections of the point $B$ onto the lines $AX,XC, CY, YA$ also lie on the same circle.

Let ${ABC}$ be an acute angled triangle, and ${H}$ a point in its interior. Let the reflections of ${H}$ through the sides ${AB}$ and ${AC}$ be called ${H_{c} }$ and ${H_{b} }$ , respectively, and let the reflections of H through the midpoints of these same sidesbe called ${H_{c}^{'} }$ and ${H_{b}^{'} }$, respectively. Show that the four points ${H_{b}, H_{b}^{'} , H_{c}}$, and ${H_{c}^{'} }$ are concyclic if and only if at least two of them coincide or ${H}$ lies on the altitude from ${A}$ in triangle ${ABC}$.

Let $BB_1$ be the symmedian of a nonisosceles acute-angled triangle $ABC$. Ray $BB_1$ meets the circumcircle of $ABC$ for the second time at point $L$. Let $AH_A, BH_B, CH_C$ be the altitudes of triangle $ABC$. Ray $BH_B$ meets the circumcircle of $ABC$ for the second time at point $T$. Prove that $H_A, H_C, T, L$ are concyclic.

Given a scalene triangle $\vartriangle ABC$ with orthocenter $H$. The midpoint of $BC$ is denoted by $M$. $AH$ intersects the circumcircle at $D \ne A$ and $DM$ intersects circumcircle of $\vartriangle ABC$ at $T\ne D$. Now, assume the reflection points of $M$ with respect to $AB,AC,AH$ are $F,E,S$. Show that the midpoints of $BE,CF,AM,TS$ are concyclic. [img]https://3.bp.blogspot.com/-v7D_A66nlD0/XnYNJussW9I/AAAAAAAALeQ/q6DMQ7w6QtI5vLwBcKqp4010c3XTCj3BgCK4BGAYYCw/s1600/imoc2019g2.png[/img]

The center $O$ of the circle $\omega$ passing through the vertex $C$ of the isosceles triangle $ABC$ ($AB = AC$) is the interior point of the triangle $ABC$. This circle intersects segments $BC$ and $AC$ at points $D \ne C$ and $E \ne C$, respectively, and the circumscribed circle $\Omega$ of the triangle $AEO$ at the point $F \ne E$. Prove that the center of the circumcircle of the triangle $BDF$ lies on the circle $\Omega$.

Let $ABC$ be a triangle and $D$ a point on the side $BC$. Point $E$ is the symmetric of $D$ with respect to $AB$. Point $F$ is the symmetric of $E$ with respect to $AC$. Point $P$ is the intersection of line $DF$ with line $AC$. Prove that the quadrilateral $AEDP$ is cyclic. (Malik Talbi)

Let $\vartriangle ABC$ be an acute triangle with altitudes $AD$, $BE$, $CF$ and orthocenter $H$. Circle $\odot V$ is the circumcircle of $\vartriangle DE F$. Let segments $FD$, $BH$ intersect at point $P$. Let segments $ED$, $HC$ intersect at point $Q$. Let $K$ be a point on $AC$ such that $VK \perp CF$. a) Prove that $\vartriangle PQH \sim \vartriangle AKV$. b) Let line $PQ$ intersect $\odot V$ at points $I,G$. Prove that points $B,I,H,G,C$ are concyclic [hide]with center the symmetric point $X$ of circumcenter $O$ of $\vartriangle ABC$ wrt $BC$.[/hide] [hide=PS.] There is a chance that those in the hide were not wanted in the problem, as I tried to understand the wording from a solutions' video. I couldn't find the original wording pdf or picture.[/hide] [img]https://cdn.artofproblemsolving.com/attachments/c/3/0b934c5756461ff854d38f51ef4f76d55cbd95.png[/img] [url=https://www.geogebra.org/m/cjduebke]geogebra file[/url]

As shown in the figure, $AB$ is the diameter of circle $\odot O$, and chords $AC$ and $BD$ intersect at point $E$, $EF\perp AB$ intersects at point $F$, and $FC$ intersects $BD$ at point $G$. Point $M$ lies on $AB$ such that $MD=MG$ . Prove that points $F$, $M$, $D$, $G$ lies on a circle. [img]https://cdn.artofproblemsolving.com/attachments/2/3/614ef5b9e8c8b16a29b8b960290ef9d7297529.jpg[/img]

The center $O$ of the circle $\omega$ passing through the vertex $C$ of the isosceles triangle $ABC$ ($AB = AC$) is the interior point of the triangle $ABC$. This circle intersects segments $BC$ and $AC$ at points $D \ne C$ and $E \ne C$, respectively, and the circumscribed circle $\Omega$ of the triangle $AEO$ at the point $F \ne E$. Prove that the center of the circumcircle of the triangle $BDF$ lies on the circle $\Omega$.

In triangle $ABC$, on side $AC$ there are points $D$ and $E$, that $AB = AD$ and $BE = EC$ ($E$ between $A$ and $D$). Point $F$ is midpoint of arc $BC$ of circumcircle of triangle $ABC$. Prove that the points $B, E, D, F$ lie on the same circle.

$ABC$ is an isosceles triangle. The base $BC$ is $1$ cm long, and legs $AB$ and $AC$ are $2$ cm long. Let the midpoint of $AB$ be $F$, and the midpoint of $AC$ be $G$. Additionally, $k$ is a circle, that is tangent to $AB$ and A$C$, and it’s points of tangency are $F$ and $G$ accordingly. Prove, that the intersection of $CF$ and $BG$ falls on the circle $k$.

Triangle $ABC$ is inscribed in a circle with center $O$. Let $D$ and $E$ be points on the respective sides $AB$ and $AC$ so that $DE$ is perpendicular to $AO$. Show that the four points $B,D,E$ and $C$ lie on a circle.

Let $ ABC $ be an inscribed triangle in circle $ (O) $. Let $ D $ be the intersection of the two tangent lines of $ (O) $ at $ B $ and $ C $. The circle passing through $ A $ and tangent to $ BC $ at $ B $ intersects the median passing $ A $ of the triangle $ ABC $ at $ G $. Lines $ BG, CG $ intersect $ CD, BD $ at $ E, F $ respectively. a) The line passing through the midpoint of $ BE $ and $ CF $ cuts $ BF, CE $ at $ M, N $ respectively. Prove that the points $ A, D, M, N $ belong to the same circle. b) Let $ AD, AG $ intersect the circumcircle of the triangles $ DBC, GBC $ at $ H, K $ respectively. The perpendicular bisectors of $ HK, HE$, and $HF $ cut $ BC, CA$, and $AB $ at $ R, P$, and $Q $ respectively. Prove that the points $ R, P$, and $Q $ are collinear.

A tetrahedron $ABCD$ is give. A line $\ell$ meets the planes $ABC,BCD,CDA,DAB$ at points $D_0,A_0,B_0,C_0$ respectively. Let $P$ be an arbitrary point not lying on $\ell$ and the planes of the faces, and $A_1,B_1,C_1,D_1$ be the second common points of lines $PA_0,PB_0,PC_0,PD_0$ with the spheres $PBCD,PCDA,PDAB,PABC$ respectively. Prove $P,A_1,B_1,C_1,D_1$ lie on a circle.

Let $\Gamma_1$ and $\Gamma_2$ be circles intersecting at points $A$ and $B$. A line through $A$ intersects $\Gamma_1$ and $\Gamma_2$ at $C$ and $D$ respectively. Let $P$ be the intersection of the lines tangent to $\Gamma_1$ at $A$ and $C$, and let $Q$ be the intersection of the lines tangent to $\Gamma_2$ at $A$ and $D$. Let $X$ be the second intersection point of the circumcircles of $BCP$ and $BDQ$, and let $Y$ be the intersection of lines $AB$ and $PQ$. Prove that $C$, $D$, $X$ and $Y$ are concyclic. [i]Proposed by Ariel García[/i]