Found problems: 254
Let point $T$ be on side $AB$ of $\Delta ABC$ be such that $AT-BT=AC-BC$. The perpendicular from point $T$ to $AB$ intersects $AC$ in point $E$ and the angle bisectors of $\angle B$ and $\angle C$ intersect the circumscribed circle $k$ of $ABC$ in points $M$ and $L$. If $P$ is the second intersection point of the line $ME$ with $k$, then prove that $P,T,L$ are collinear.
Points $M$ and $N$ are marked on the straight line containing the side $AC$ of triangle $ABC$ so that $MA = AB$ and $NC = CB$ (the order of the points on the line: $M, A, C, N$). Prove that the center of the circle inscribed in triangle $ABC$ lies on the common chord of the circles circumscribed around triangles $MCB$ and $NAB$ .
Given points $A, B, C$ on a line, equilateral triangles $ABC_1$ and $BCA_1$ constructed on segments $AB$ and $BC$, and midpoints $M$ and $N$ of $AA_1$ and $CC_1$, respectively. Prove that $\vartriangle BMN$ is equilateral. (We assume that $B$ lies between $A$ and $C$, and points $A_1$ and $C_1$ lie on the same side of line $AB$)
Let $ABCD$ be a quadrilateral and $P$ a point in the plane of the quadrilateral. Let $M,N$ be on the sides $AC,BD$ respectively such that $PM \parallel BC, PN \parallel AD$. $AC$ meets $BD$ at $E$. Prove that the orthocenter of triangles $EBC, EAD, EMN$ are collinear if and only if $P$ is on the line $AB$.
Đỗ Thanh Sơn
PS. Instead of the word [b]collinear[/b], it was written [b]concurrent[/b], probably a typo.
Two circles $w_1$ and $w_2$ intersect at points $A$ and $B$. Tangents $\ell_1$ and $\ell_2$ respectively are drawn to them through point $A$. The perpendiculars dropped from point $B$ to $\ell_2$ and $\ell_1$ intersects the circles $w_1$ and $w_2$, respectively, at points $K$ and $N$. Prove that points $K, A$ and $N$ lie on one straight line.
In the triangle $ABC$ the relationship $AB+AC = 2BC$ holds. Let $I$ and $M$ be the incenter and intersection point of the medians of triangle $ABC$ respectively, $AL$ its angle bisector, and point $P$ the orthocenter of triangle $BIC$. Prove that the points $L, M, P$ lie on a straight line.
(Matvii Kurskyi)
In triangle $ABC$, the altitude through $B$ intersects $AC$ at $E$ and the altitude through $C$ intersects $AB$ at $F$. Point $T$ is such that $AETF$ is a parallelogram and points $ A$ ,$T$ lie on different half-planes wrt the line $EF$. Point $D$ is such that $ABDC$ is a parallelogram and points $ A$ ,$D$ lie in different half-planes wrt line $BC$. Prove that $T, D$ and the orthocenter of $ABC$ are collinear.
Let $ABC$ be a triangle with circumcircle $\omega$ . Point $D$ lies on the arc $BC$ of $\omega$ and is different than $B,C$ and the midpoint of arc $BC$. Tangent of $\Gamma$ at $D$ intersects lines $BC$, $CA$, $AB$ at $A',B',C'$, respectively. Lines $BB'$ and $CC'$ intersect at $E$. Line $AA'$ intersects the circle $\omega$ again at $F$. Prove that points $D,E,F$ are collinear.
(Saudi Arabia)
Two congruent equilateral triangles $ABC$ and $A'B'C'$ in the plane are given. Show that the midpoints of the segments $AA',BB', CC'$ either are collinear or form an equilateral triangle.
Let $ABCD$ be a quadrilateral inscribed in a circle $(O)$. Let $(O_1), (O_2), (O_3), (O_4)$ be the circles going through $(A,B), (B,C),(C,D),(D,A)$. Let $X, Y,Z, T$ be the second intersection of the pairs of the circles: $(O_1)$ and $(O_2), (O_2)$ and $(O_3), (O_3)$ and $(O_4), (O_4)$ and $(O_1)$.
(a) Prove that $X, Y,Z, T$ are on the same circle of radius $I$.
(b) Prove that the midpoints of the line segments $O_1O_3,O_2O_4,OI$ are collinear.
Nguyễn Văn Linh
Let $I$ be the centre of the inscribed circle of the non-isosceles triangle $ABC$, and let the circle touch the sides $BC, CA, AB$ at the points $A_1, B_1, C_1$ respectively. Prove that the centres of the circumcircles of $\vartriangle AIA_1,\vartriangle BIB_1$ and $\vartriangle CIC_1$ are collinear.
Let $ABC$ be an acute-angled and non-isosceles triangle with orthocenter $H$. Let $O$ be the center of the circumscribed circle of triangle $ABC$ and let $K$ be center of the circumscribed circle of triangle $AHO$. Prove that the reflection of $K$ wrt $OH$ lies on $BC$.
Given a quadrilateral $ABCD$, around which you can circumscribe a circle. The perpendicular bisectors of sides $AD$ and $CD$ intersect at point $Q$ and intersect sides $BC$ and $AB$ at points $P$ and $K$ resepctively. It turned out that the points $K, B, P, Q$ lie on the same circle. Prove that the points $A, Q, C$ lie on one line.
(Olena Artemchuk)
Given a triangle $ABC$ and an arbitrary point $D$.The lines passing through $D$ and perpendicular to segments $DA, DB, DC$ meet lines $BC, AC, AB$ at points $A_1, B_1, C_1$ respectively. Prove that the midpoints of segments $AA_1, BB_1, CC_1$ are collinear.
About the triangle $ABC$ it is known that $AM$ is its median, and $\angle AMC = \angle BAC$. On the ray $AM$ lies the point $K$ such that $\angle ACK = \angle BAC$. Prove that the centers of the circumcircles of the triangles $ABC, ABM$ and $KCM$ lie on the same line.
A circle inscribed in triangle $ABC$ has center $O$ and touches side $AC$ at point $K$. A second circle also has center $O$, intersects all sides of triangle $ABC$. Let $E$ and $F$ be the corresponding points of intersection with sides $AB$ and $BC$, closest to vertex $B$; $B_1$ and $B_2$ are the points of its intersection with side $AC$, and $B_1$ is closer to $A$. Prove that points $B$, $K$ and point $P$, the intersections of the segments $B_2E$ and $B_1F$ lie on the same straight line.
$3n$ points are given ($n\ge 1$) in the plane, each $3$ of them are not collinear. Prove that there are $n$ distinct triangles with the vertices those points.
Let $I$ be the center of a circle inscribed in triangle $ABC$, in which $\angle BAC = 60 ^o$ and $AB \ne AC$. The points $D$ and $E$ were marked on the rays $BA$ and $CA$ so that $BD = CE = BC$. Prove that the line $DE$ passes through the point $I$.
Let $ABCD$ be a quadrilateral, let $O$ be the intersection point of diagonals $AC$ and $BD$, and let $P$ be the intersection point of sides $AB$ and $CD$. Consider the parallelograms $AODE$ and $BOCF$. Prove that $E, F$ and $P$ are collinear.
Let $BB_1$ and $CC_1$ be the altitudes of an acute-angled triangle $ABC$, which intersect its angle bisector $AL$ at two different points $P$ and $Q$, respectively. Denote by $F$ such a point that $PF\parallel AB$ and $QF\parallel AC$, and by $T$ the intersection point of the tangents drawn at points $B$ and $C$ to the circumscribed circle of the triangle $ABC$. Prove that the points $A, F$ and $T$ lie on the same line.
Let $H$ be an orthocenter of an acute triangle $ABC$. Prove that midpoints of $AB$ and $CH$ and intersection point of angle bisectors of $\angle CAH$ and $\angle CBH$ lie on the same line.
Let $N$ be the set of natural numbers, and let $f: N \to N$ be a function satisfying $f(x) + f(x + 2) < 2 f(x + 1)$ for any $x \in N$. Prove that there exists a straight line in the $xy$-plane which contains infinitely many points with coordinates $(n,f(n))$.
Let P be an arbitrary point on the arc $AC$ of the circumcircle of a fixed triangle $ABC$, not containing $B$. The bisector of angle $APB$ meets the bisector of angle $BAC$ at point $P_a$ the bisector of angle $CPB$ meets the bisector of angle $BCA$ at point $P_c$. Prove that for all points $P$, the circumcenters of triangles $PP_aP_c$ are collinear.
by I. Dmitriev
On the circle with diameter $AB$, the point $M$ was selected and fixed. Then the point ${{Q} _ {i}}$ is selected, for which the chord $M {{Q} _ {i}}$ intersects $AB$ at the point ${{K} _ {i}}$ and thus $ \angle M {{K} _ {i}} B <90 {} ^ \circ$. A chord that is perpendicular to $AB$ and passes through the point ${{K} _ {i}}$ intersects the line $B {{Q} _ {i}}$ at the point ${{P } _ {i}}$. Prove that the points ${{P} _ {i}}$ in all possible choices of the point ${{Q} _ {i}}$ lie on the same line.
(Igor Nagel)
Given an acute $\vartriangle ABC$ with orthocenter $H$. Let $M_a$ be the midpoint of $BC. M_aH$ intersects the circumcircle of $\vartriangle ABC$ at $X_a$ and $AX_a$ intersects $BC$ at $Y_a$. Define $Y_b, Y_c$ in a similar way. Prove that $Y_a, Y_b,Y_c$ are collinear.
[img]https://2.bp.blogspot.com/-yjISBHtRa0s/XnSKTrhhczI/AAAAAAAALds/e_rvs9glp60L1DastlvT0pRFyP7GnJnCwCK4BGAYYCw/s320/imoc2017%2Bg4.png[/img]