Found problems: 478
Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.
Let $ABC$ be a triangle and $P$ be a point in the plane of the triangle. The lines $AP,BP, CP$ meets $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Let $A_2,B_2,C_2$ be the Miquel point of the complete quadrilaterals $AB_1PC_1BC$, $BC_1PA_1CA$, $CA_1PB_1AB$. Prove that the circumcircles of the triangles $APA_2$,$BPB_2$, $CPC_2$, $BA_2C$, $AB_2C$, $AC_2B$ have a point of concurrency.
Nguyễn Văn Linh
Let $ABC$ be an acute triangle. Define $A_1$ the midpoint of the largest arc $BC$ of the circumcircle of $ABC$ . Let $A_2$ and $A_3$ be the feet of the perpendiculars from $A_1$ on the lines $AB$ and $AC$ , respectively. Define $B_1$, $B_2$, $B_3$, $C_1$, $C_2$, and $C_3$ analogously. Show that the lines $A_2A_3$, $B_2B_3$, $C_2C_3$ are concurrent.
A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.
$\vartriangle ABC$ is a triangle with $AB < BC, \Gamma$ its circumcircle, $K$ the midpoint of the minor arc $CA$ of the circle $C$ and $T$ a point on $\Gamma$ such that $KT$ is perpendicular to $BC$. If $A',B'$ are the intouch points of the incircle of $\vartriangle ABC$ with the sides $BC,AC$, prove that the lines $AT,BK,A'B'$ are concurrent.
Consider a regular polygon $A_0A_1...A_{n-1}, n \ge 3$, and $m \in\{1, 2, ..., n - 1\}, m \ne n/2$. For any number $i \in \{0,1, ... , n - 1\}$, let $r(i)$ be the remainder of $i + m$ at the division by $n$. Prove that no three segments $A_iA_{r(i)}$ are concurrent.
In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$, point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$. The excircle opposite to $A$ touches $\overline{BC}$ at point $E$. Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$. Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$.
Lines containing the medians of the triangle $ABC$ intersect its circumscribed circle for a second time at the points $A_1, B_1, C_1$. The straight lines passing through $A,B,C$ parallel to opposite sides intersect it at points $A_2, B_2, C_2$. Prove that lines $A_1A_2,B_1B_2,C_1C_2$ intersect at one point.
The inscribed circle of a triangle $ABC$ has the center $O$ and touches the triangle sides $BC, CA$ and $AB$ at points $X, Y$ and $Z$, respectively. The parallels to the straight lines $ZX, XY$ and $YZ$ the straight lines $BC, CA$ and $AB$ (in this order!) intersect through the point $O$. Points $K, L$ and $M$. Then the parallels to the straight lines $CA, AB$ and $BC$ intersect through the points $K, L$ and $M$ in one point.
Let $ABC$ be a triangle inscribed in a circle $(O)$. Let $P$ be an arbitrary point in the plane of triangle $ABC$. Points $A',B',C'$ are the reflections of $P$ about the lines $BC,CA,AB$ respectively. $X$ is the intersection, distinct from $A$, of the circle with diameter $AP$ and the circumcircle of triangle $AB'C'$. Points $Y,Z$ are defined in the same way. Prove that five circles $(O), (AB'C')$, $(BC'A'), (CA'B'), (XY Z)$ have a point in common.
Nguyễn Văn Linh
Let be a triangle $ ABC $, the midpoint of the $ AC $ and $ N $ side, and the midpoint of the $ AB $ side. Let $ r $ and $ s $ reflect the straight lines $ BM $ and $ CN $ on the straight $ BC $, respectively. Also define $ D $ and $ E $ as the intersection of the lines $ r $ and $ s $ and the line $ MN $, respectively. Let $ X $ and $ Y $ be the intersection points between the circumcircles of the triangles $ BDM $ and $ CEN $, $ Z $ the intersection of the lines $ BE $ and $ CD $ and $ W $ the intersection between the lines $ r $ and $ s $. Prove that $ XY, WZ $, and $ BC $ are concurrents.
Points $A', B', C'$ are the feet of the altitudes $AA', BB'$ and $CC'$ of an acute triangle $ABC$. A circle with center $B$ and radius $BB'$ meets line $A'C'$ at points $K$ and $L$ (points $K$ and $A$ are on the same side of line $BB'$). Prove that the intersection point of lines $AK$ and $CL$ belongs to line $BO$ ($O$ is the circumcenter of triangle $ABC$).
Let$ AB$ and $AC$ be the tangents from a point $A$ to a circle $ \Omega$. Let $M$ be the midpoint of $BC$ and $P$ be an arbitrary point on this segment. A line $AP$ meets $ \Omega$ at points $D$ and $E$. Prove that the common external tangents to circles $MDP$ and $MPE$ meet on the midline of triangle $ABC$.
Let $ABCDEF$ be a convex hexagon satisfying $AC = DF, CE = FB$ and $EA = BD$. Prove that the lines connecting the midpoints of opposite sides of the hexagon $ABCDEF$ intersect in one point.
Let $ABCD$ be a trapezoid, where $AB$ and $CD$ are parallel. Let $P$ be a point on the side $BC$. Show that the parallels to $AP$ and $PD$ intersect through $C$ and $B$ to $DA$, respectively.
Let $ABC$ be an acute angled triangle. The arc between $A$ and $B$ of the circumcircle of $ABC$ is reflected through the line $AB$, and the arc between $A$ and $C$ of the circumcircle of $ABC$ is reflected over the line $AC$. Obviously these two reflected arcs intersect at the point $A$. Prove that they also intersect at another point inside the triangle $ABC$.
The circles ${{w} _ {1}}$ and ${{w} _ {2}}$ intersect at points $P$ and $Q$. Let $AB$ and $CD$ be parallel diameters of circles ${ {w} _ {1}}$ and ${{w} _ {2}} $, respectively. In this case, none of the points $A, B, C, D$ coincides with either $P$ or $Q$, and the points lie on the circles in the following order: $A, B, P, Q$ on the circle ${{w} _ {1} }$ and $C, D, P, Q$ on the circle ${{w} _ {2}} $. The lines $AP$ and $BQ$ intersect at the point $X$, and the lines $CP$ and $DQ$ intersect at the point $Y, X \ne Y$. Prove that all lines $XY$ for different diameters $AB$ and $CD$ pass through the same point or are all parallel.
(Serdyuk Nazar)
We are given $n$ circles which have the same center. Two lines $D_1,D_2$ are concurent in $P$, a point inside all circles. The rays determined by $P$ on the line $D_i$ meet the circles in points $A_1,A_2,...,A_n$ and $A'_1, A'_2,..., A'_n$ respectively and the rays on $D_2$ meet the circles at points $B_1,B_2, ... ,B_n$ and $B'_2, B'_2 ..., B'_n$ (points with the same indices lie on the same circle). Prove that if the arcs $A_1B_1$ and $A_2B_2$ are equal then the arcs $A_iB_i$ and $A'_iB'_i$ are equal, for all $i = 1,2,... n$.
On the sides $AB, BC, CA$ of triangle $ABC$, points $C', A', B'$ are taken.
Prove that for the areas of the corresponding triangles, the inequality holds:
$$S_{ABC}S^2_{A'B'C'}\ge 4S_{AB'C'}S_{BC'A'}S_{CA'B'}$$
and equality is achieved if and only if the lines $AA', BB', CC'$ intersect at one point.
$A_1, B_1, C_1$ are the midpoints of the sides $BC,CA,BA$ respectively of an equilateral triangle $ABC$. Three parallel lines, passing through $A_1, B_1, C_1$ intersect, respectively, lines $B_1C_1, C_1A_1, A_1B_1$ at points $A_2, B_2, C_2$. Prove that the lines $AA_2, BB_2, CC_2$ intersect at one point lying on the circle circumscribed around the triangle $ABC$.
Consider the triangle $ABC$ with circumcenter $O$. Let $D$ be the intersection of the angle bisector of $\angle{A}$ with $BC$. Show that $OA$, the perpendicular bisector of $AD$ and the perpendicular to $BC$ passing through $D$ are concurrent.
A line $l$ is drawn through the intersection point $H$ of altitudes of acute-angle triangles. Prove that symmetric images $l_a, l_b, l_c$ of $l$ with respect to the sides $BC,CA,AB$ have one point in common, which lies on the circumcircle of $ABC.$
In triangle $ABC$, let $I_a$ $,I_b$, $I_c$ be the centers of the excircles tangent to sides $BC$, $CA$, $AB$, respectively. Let $P$ and $Q$ be the tangency points of the excircle of center $I_a$ with lines $AB$ and $AC$. Line $PQ$ intersects $I_aB$ and $I_aC$ at $D$ and $E$. Let $A_1$ be the intersection of $DC$ and $BE$. In an analogous way we define points $B_1$ and $C_1$. Prove that $AA_1$, $BB_1$ , $CC_1$ are concurrent.
The altitude $AA'$, the median $BB'$, and the angle bisector $CC'$ of a triangle $ABC$ are concurrent at point $K$. Given that $A'K = B'K$, prove that $C'K = A'K$.
Triangles $\vartriangle ABC$ and $\vartriangle A_1B_1C_1$ lie on different planes. Line $AB$ intersects line $A_1B_1$, line $BC$ intersects line $B_1C_1$ and line $CA$ intersects line $C_1A_1$. Prove that either the three lines $AA_1, BB_1, CC_1$ meet at one point or that they are all parallel.