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## 5 concurrent lines

Given a cyclic quadrilateral $ABCD$ and its circumcircle $(O)$. Denote $P$ the intersection of $AC$ and $BD, I_1, I_2, I_3, I_4$ the incenters of triangles $ABP, BCP, CDP, DAP$, respectively. Let $\omega_1$ be the circle which is tangent to rays $PA,PB$ and $(O)$ at $X_1$. Similarly we define $X_2,X_3,X_4$. Prove that five lines $I_1X_3, I_2X_4, I_3X_1,I_4X_2$ and $OP$ are concurrent.

## Some problems of me about coaxal circles.

Problem 1: Given triangle $ABC$ and its incircle $(I). (I)$ touches $BC, CA, AB$ at $A_1,B_1,C_1$, respectively. Let $A_2,B_2,C_2$ be the points on $IA_1,IB_1,IC_1$ such that $\frac {\overline{IA_2}}{\overline{IA_1}} = \frac {\overline{IB_2}}{\overline{IB_1}} = \frac {\overline{IC_2}}{\overline{IC_1}}$. Prove that 3 circles $(AIA_2),(BIB_2), (CIC_2)$ concur at the point $Q\neq I$.

Problem 2: Given triangle $ABC$ and its circumcenter $O$. Denote $P$ a point such that the pedal triangle $A'B'C'$ of $P$ wrt $\Delta ABC$ is the cevian triangle of point $Q$ wrt $\Delta ABC$. Prove that $(APA'),(BPB')$ and $(CPC')$ concur at a point which lies on $OP$.

Problem 3: Given triangle $ABC$ and its incircle $(I)$. Let $A_1,B_1,C_1$ be the intersections of $AI$ and $BC,BI$ and $CA,CI$ and $AB$, respectively; $A_2$ be the intersection of a line through $I$ and perpendicular to $AI$ and $BC$, similarly define $B_2, C_2$. Show that $(AA_1A_2), (BB_1B_2), (CC_1C_2)$ concur at two points.

Problem 4: Given triangle $ABC$ and its circumcircle $(O)$. Denote $M_a, M_b, M_c$ the midpoints of $BC, CA, AB$, respectively, $P$ an arbitrary point inside $\Delta ABC, A',B',C'$ the second intersections of $AP, BP, CP$ and $(O)$. Prove that $(A'OM_a), (B'OM_b), (C'OM_c)$ concur at two points.

Problem 5: Given triangle $ABC$, its incircle $(I)$ and its circumcircle $(O)$. Denote $A_1$ be the intersection of the line through  $I$ and perpendicular to $AI$ and $BC$, similarly define $B_1,C_1$.  Denote $A_2,B_2,C_2$ the projections of $I$ on $BC,CA,AB$, respectively. Prove that three circles $(AA_1A_2), (BB_1B_2), (CC_1C_2)$ are coaxal.

Problem 6: Given a triangle $ABC$ with its circumcenter $O$. Let $d$ be an arbitrary line in the plane. $d$ intersects $BC, CA, AB$ at $X, Y, Z$, respectively. Let $H$ be the projection of $O$ on $d$. Prove that 3 circles $(AXH), (BYH), (CZH)$ are coaxal.

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=366220

## Near Hagge circle 2

Given triangle $ABC$ , its orthocenter $H$ and its circumcircle $(O)$. Denote $A_1,B_1,C_1$ the projections of $A,B,C$ on $BC,CA,AB$, respectively; $P$ an arbitrary point on $OH. AP,BP,CP$ intersect $(O)$ again at $A_2,B_2,C_2$, respectively; $A_3,B_3,C_3$ the reflections of $A_2,B_2,C_2$ wrt $A_1,B_1,C_1$, respectively. Prove that 4 points $H,A_3,B_3,C_3$ are concyclic and its circumcenter lies on $OH$.

The solution of the generalization can be found at here

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=345850&start=0&sid=ac77486215e4ffc3e51335237b443f73

Given triangle $ABC$ and its orthocenter $H$, its circumcircle $(O)$. Let $P$ be an arbitrary point on the plane. $AP,BP,CP$ intersects $(O)$ again at $A_1,B_1,C_1$, respectively. Let $A_2B_2C_2$ be the pedal triangle of $P$ wrt $\Delta ABC, A_3,B_3,C_3$ be the reflections of $A_1,B_1,C_1$ wrt $A_2,B_2,C_2$ ,respectively. Prove that 4 points $H,A_3,B_3,C_3$ are concyclic.