Let $A B C$ be an acut-angles triangle of incenter $I$, circumcenter $O$ and inradius $r.$ Let $\omega$ be the inscribed circle of the triangle $A B C$. $A_1$ is the point of $\omega$ such that $A IA_1O$ is a convex trapezoid of bases $A O$ and $IA_1$. Let $\omega_1$ be the circle of radius $r$ which goes through $A_1$, tangent to the line $A B$ and is different from $\omega$ . Let $\omega_2$ be the circle of radius $r$ which goes through $A_1$, is tangent to the line $A C$ and is different from $\omega$ . Circumferences $\omega_1$ and $\omega_2$ they are cut at points $A_1$ and $A_2$. Similarly are defined points $B_2$ and $C_2$. Prove that the lines $A A_2, B B_2$ and $CC2$ they are concurrent.

A convex quadrilateral $ABCD$ is inscribed into a circle $\omega$ . Suppose that there is a point $X$ on the segment $AC$ such that the $XB$ and $XD$ tangents to the circle $\omega$ . Tangent of $\omega$ at $C$, intersect $XD$ at $Q$. Let $E$ ($E\ne A$) be the intersection of the line $AQ$ with $\omega$ . Prove that $AD, BE$, and $CQ$ are concurrent.

The squares $OABC$ and $OA_1B_1C_1$ are situated in the same plane and are directly oriented. Prove that the lines $AA_1$ , $BB_1$, and $CC_1$ are concurrent.

Let $ABC$ be a triangle and let $L$, $M$, $N$ be the midpoints of the sides $BC$, $CA$ and $AB$ , respectively. The points $P$ and $Q$ lie on $AB$ and $BC$, respectively; the points $R$ and $S$ are such that $N$ is the midpoint of $PR$ and $L$ is the midpoint of $QS$. Show that if $PS$ and $QR$ are perpendicular, then their intersection lies on in the circumcircle of triangle $LMN$.

Circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects the circle $k_1$ at points $A$ and $C$, the circle $K_2$ at points $B$ and $D$ so that the points $A,B,C$ and $D$ lie on the line $\ell$ are in that order. Let $X$ a point on the line $MN$ such that the point $M$ is located between the points $X$ and $N$. Let $P$ be the intersection of lines $AX$ and $BM$, and $Q$ be the intersection of lines $DX$ and $CM$. If $K$ is the midpoint of segment $AD$ and $L$ is the midpoint of segment $BC$, prove that the lines $XK$ and $ML$ intersect on the line $PQ$.

Let $p, q$ and $r$ be three concurrent lines in the plane such that the angle between any two of them is $60^o$. Let $a$, $b$ and $c$ be real numbers such that $0 < a \le b \le c$. (a) Prove that the set of points whose distances from $p, q$ and $r$ are respectively less than $a, b$ and $c$ consists of the interior of a hexagon if and only if $a + b > c$. (b) Determine the length of the perimeter of this hexagon when $a + b > c$.

Check and justify , if in every tetrahedron are concurrent: a) The perpendiculars to the faces at their circumcenters. b) The perpendiculars to the faces at their orthocenters. c) The perpendiculars to the faces at their incenters. If so, characterize with some simple geometric property the point in that attend If not, show an example that clearly shows the not concurrency.

In fig. the bisectors of the angles $\angle DAC$, $ \angle EBD$, $\angle ACE$, $\angle BDA$ and $\angle CEB$ intersect at one point. Prove that the bisectors of the angles $\angle TPQ$, $\angle PQR$, $\angle QRS$, $\angle RST$ and $\angle STP$ also intersect at one point. [img]https://cdn.artofproblemsolving.com/attachments/6/e/870e4f20bc7fdcb37534f04541c45b1cd5034a.png[/img]

Given a triangle $ABC$ and a point $K$ . The lines $AK$,$BK$,$CK$ hit the opposite side of the triangle at $D,E,F$ respectively. On the exterior of $ABC$, we construct three pairs of similar triangles: $BDM$,$DCN$ on $BD$,$DC$, $CEP$,$EAQ$ on $CE$,$EA$, and $AFR$,$FBS$ on $AF$, $FB$. The lines $MN$,$PQ$,$RS$ intersect each other form a triangle $XYZ$. Prove that $AX$,$BY$,$CZ$ are concurrent.

Let $A, B, C, A', B', C'$ be six pairs of different points. Prove that the Circles $BCA'$, $CAB'$ and $ABC'$ have a common point, then the Circles $B'C'A, C'A'B$ and $A'B'C$ also share a common point. Note: For three pairs of different points $X, Y$ and $Z$ we define the [i]Circle [/i] $XYZ$ as the circumcircle of the triangle $XYZ$, or - in the case when the points $X, Y$ and $Z$ lie on a straight line - this straight line.

Let $ABCD$ be a tangential quadrilateral with $BC> BA$. The point $P$ is on the segment $BC$, such that $BP = BA$ . Show that the bisector of $\angle BCD$, the perpendicular on line $BC$ through $P$ and the perpendicular on $BD$ through $A$, intersect at one point.

A non-isosceles trapezoid $ABCD$ ($AB // CD$) is given. An arbitrary circle passing through points $A$ and $B$ intersects the sides of the trapezoid at points $P$ and $Q$, and the intersect the diagonals at points $M$ and $N$. Prove that the lines $PQ, MN$ and $CD$ are concurrent.

Let $ABCD$ be a parallelogram with vertices in order clockwise and let $E$ be the intersection of its diagonals. The circle of diameter $DE$ intersects the segment $AD$ at $L$ and $EC$ at $H$. The circumscribed circle of $LEB$ intersects the segment $BC$ at $O$. Prove that the lines $HD$ , $LE$ and $BC$ are concurrent if and only if $EO = EC$.

Let $E_1, E_2$, and $E_3$ be three mutually intersecting ellipses, all in the same plane. Their foci are respectively $F_2, F_3; F_3, F_1$; and $F_1, F_2$. The three foci are not on a straight line. Prove that the common chords of each pair of ellipses are concurrent.

Let $H$ be the orthocenter of triangle $ABC$, $X$ be an arbitrary point. A circle with a diameter of $XH$ intersects lines $AH, BH, CH$ at points $A_1, B_1, C_1$ for the second time, and lines $AX BX, CX$ at points $A_2, B_2, C_2$. Prove that lines A$_1A_2, B_1B_2, C_1C_2$ intersect at one point.

Let $ABC$ be a scalene triangle, and $A_o$, $B_o,$ $C_o$ be the midpoints of $BC$, $CA$, $AB$ respectively. The bisector of angle $C$ meets $A_oCo$ and $B_oC_o$ at points $B_1$ and $A_1$ respectively. Prove that the lines $AB_1$, $BA_1$ and $A_oB_o$ concur.

Given a triangle $ABC$ inscribed in circle $c(O,R)$ (with center $O$ and radius $R$) with $AB<AC<BC$ and let $BD$ be a diameter of the circle $c$. The perpendicular bisector of $BD$ intersects line $AC$ at point $M$ and line $AB$ at point $N$. Line $ND$ intersects the circle $c$ at point $T$. Let $S$ be the second intersection point of cicumcircles $c_1$ of triangle $OCM$, and $c_2$ of triangle $OAD$. Prove that lines $AD, CT$ and $OS$ pass through the same point.

Points $A, B, C$ and $D$ lie on a circle, in that order clockwise, such that there is a point $E$ on segment $CD$ with the property that $AD = DE$ and $BC = EC$. Prove that the intersection point of the bisectors of the angles $\angle DAB$ and $\angle ABC$ is on the line $CD$.

Let $BM$ be a median of nonisosceles right-angled triangle $ABC$ ($\angle B = 90^o$), and $Ha, Hc$ be the orthocenters of triangles $ABM, CBM$ respectively. Prove that lines $AH_c$ and $CH_a$ meet on the medial line of triangle $ABC$. (D. Svhetsov)

Let $ABC$ be an acute nonisosceles triangle with incenter $I$ and $(d)$ is an arbitrary line tangent to $(I)$ at $K$. The lines passes through $I$, perpendicular to $IA, IB, IC$ cut $(d)$ at $A_1, B_1,C_1$ respectively. Suppose that $(d)$ cuts $BC, CA, AB$ at $M,N, P$ respectively. The lines through $M,N,P$ and respectively parallel to the internal bisectors of $A, B, C$ in triangle $ABC$ meet each other to define a triange $XYZ$. Prove that three lines $AA_1, BB_1, CC_1$ are concurrent and $IK$ is tangent to the circle $(XY Z)$

As shown in figure , the inscribed circle of $ABC$ is intersects $BC$, $CA$, $AB$ at points $D$, $E$, $F$, repectively, and $P$ is a point inside the inscribed circle. The line segments $PA$, $PB$ and $PC$ intersect respectively the inscribed circle at points $X$, $Y$ and $Z$. Prove that the three lines $XD$, $YE$ and $ZF$ have a common point. [img]https://cdn.artofproblemsolving.com/attachments/e/9/bbfb0394b9db7aa5fb1e9a869134f0bca372c1.png[/img]

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$ respectively such that \[ OD + DH = OE + EH = OF + FH\] and the lines $AD$, $BE$, and $CF$ are concurrent.

Inside a circle $c$ there are circles $c_1, c_2$ and $c_3$ which are tangent to $c$ at points $A, B$ and $C$ correspondingly, which are all different. Circles $c_2$ and $c_3$ have a common point $K$ in the segment $BC$, circles $c_3$ and $c_1$ have a common point $L$ in the segment $CA$, and circles $c_1$ and $c_2$ have a common point $M$ in the segment $AB$. Prove that the circles $c_1, c_2$ and $c_3$ intersect in the center of the circle $c$.

Consider $n$ different points lying on a circle, such that there are not three chords defined by that point that pass through the same interior point of the circle. a) Find the value of $n$, if the numbers of triangles that are defined using $3$ of the n points is equal to $2n$ b) Find the value of $n$, if the numbers of the intersection points of the chords that are interior to the circle is equal to $5n$.