Found problems: 254
Let $ABC$ be an acute triangle, not isoceles triangle and $(O), (I)$ be its circumcircle and incircle respectively. Let $A_1$ be the the intersection of the radical axis of $(O), (I)$ and the line $BC$. Let $A_2$ be the point of tangency (not on $BC$) of the tangent from $A_1$ to $(I)$. Points $B_1,B_2,C_1,C_2$ are defined in the same manner. Prove that
(a) the lines $AA_2,BB_2,CC_2$ are concurrent.
(b) the radical centers circles through triangles $BCA_2, CAB_2$ and $ABC_2$ are all on the line $OI$.
Lê Phúc Lữ
Let $AA _1$ and $BB_1$ be the altitudes of an isosceles acute-angled triangle $ABC, M$ the midpoint of $AB$. The circles circumscribed around the triangles $AMA_1$ and $BMB_1$ intersect the lines $AC$ and $BC$ at points $K$ and $L$, respectively. Prove that $K, M$, and $L$ lie on the same line.
Consider the right isosceles $\vartriangle ABC$ at $ A$. Let $L$ be the intersection of the bisector of $\angle ACB$ with $AB$ and $K$ the intersection point of $CL$ with the bisector of $BC$. Let $X$ be the point on line $AK$ such that $\angle KCX = 90^o$ and let $Y$ be the point of intersection of $CX$ with the circumcircle of $\vartriangle ABC$. Let $Y'$ the reflection of point $Y$ wrt $BC$. Prove that $B - K -Y'$.
Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.
Let $ABC$ be an acute triangle and $\Gamma$ be its circumcircle. Line $\ell$ is tangent to $\Gamma$ at $A$ and let $D$ and $E$ be distinct points on $\ell$ such that $AD = AE$. Suppose that $B$ and $D$ lie on the same side of line $AC$. The circumcircle $\Omega_1$ of $\vartriangle ABD$ meets $AC$ again at $F$. The circumcircle $\Omega_2$ of $\vartriangle ACE$ meets $AB$ again at $G$. The common chord of $\Omega_1$ and $\Omega_2$ meets $\Gamma$ again at $H$. Let $K$ be the reflection of $H$ across line $BC$ and let $L$ be the intersection of $BF$ and $CG$. Prove that $A, K$ and $L$ are collinear.
Circles $\Gamma_1$ and $\Gamma_2$ meet at points $X$ and $Y$. A circle $S_1$ touches internally $\Gamma_1$ at $A$ and $\Gamma_2$ externally at $B$. A circle $S_2$ touches $\Gamma_2$ internally at $C$ and $\Gamma_1$ externally at $D$. Prove that the points $A, B, C, D$ are either collinear or concyclic.
(A. Voidelevich)
The diagonals of the cyclic quadrilateral $ABCD$ intersect at the point $E$. Let $P$ and $Q$ are the centers of the circles circumscribed around the triangles $BCE$ and $DCE$, respectively. A straight line passing through the point $P$ parallel to $AB$, and a straight line passing through the point $Q$ parallel to $AD$, intersect at the point $R$. Prove that the point $R$ lies on segment $AC$.
Let $\Omega$ be the inscribed circle of a triangle $\vartriangle ABC$. Let $D, E$ and $F$ be the tangency points of $\Omega$ and the sides $BC, CA$ and $AB$, respectively, and let $AD, BE$ and $CF$ intersect $\Omega$ at $K, L$ and $M$, respectively, such that $D, E, F, K, L$ and $M$ are all distinct. The tangent line of $\Omega$ at $K$ intersects $EF$ at $X$, the tangent line of $\Omega$ at $L$ intersects $DE$ at $Y$ , and the tangent line of $\Omega$ at M intersects $DF$ at $Z$. Prove that $X,Y$ and $Z$ are collinear.
The line passing through the center of the equilateral triangle $ ABC $ intersects the lines $ AB $, $ BC $ and $ CA $ at the points $ {{C} _ {1}} $, $ {{A} _ {1}} $ and $ {{B} _ {1}} $, respectively. Let $ {{A} _ {2}} $ be a point that is symmetric $ {{A} _ {1}} $ with respect to the midpoint of $ BC $; the points $ {{B} _ {2}} $ and $ {{C} _ {2}} $ are defined similarly. Prove that the points $ {{A} _ {2}} $, $ {{B} _ {2}} $ and $ {{C} _ {2}} $ lie on the same line tangent to the inscribed circle of the triangle $ ABC $.
(Serdyuk Nazar)
Altitudes $AD$ and $BE$ of an acute triangle $ABC$ intersect at $H$. Let $C_1 (H,HE)$ and $C_2(B,BE)$ be two circles tangent at $AC$ at point $E$. Let $P\ne E$ be the second point of tangency of the circle $C_1 (H,HE)$ with its tangent line going through point $C$, and $Q\ne E$ be the second point of tangency of the circle $C_2(B,BE)$ with its tangent line going through point $C$. Prove that points $D, P$, and $Q$ are collinear.
Two fixed circles $\omega_1$ and $\omega_2$ pass through point $O$. A circle of an arbitrary radius $R$ centered at $O$ meets $\omega_1$ at points $A$ and $B$, and meets $\omega_2$ at points $C$ and $D$. Let $X$ be the common point of lines $AC$ and $BD$. Prove that all the points X are collinear as $R$ changes.
Let $O$ be the incenter of a tangential quadrilateral $ABCD$. Prove that the orthocenters of $\vartriangle AOB$, $\vartriangle BOC$, $\vartriangle COD$, $\vartriangle DOA$ lie on a line.
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)
Given an acute isosceles triangle $ABC, AK$ and $CN$ are its angle bisectors, $I$ is their intersection point . Let point $X$ be the other intersection point of the circles circumscribed around $\vartriangle ABC$ and $\vartriangle KBN$. Let $M$ be the midpoint of $AC$. Prove that the Euler line of $\vartriangle ABC$ is perpendicular to the line $BI$ if and only if the points $X, I$ and $M$ lie on the same line.
(Kivva Bogdan)
Let $ABCD$ be a rhombus and $P$ be a point on its side $BC$. The circle passing through $A, B$, and $P$ intersects $BD$ once more at the point $Q$ and the circle passing through $C,P$ and $Q$ intersects $BD$ once more at the point $R$. Prove that $A, R$ and $P$ lie on the one straight line.
(D. Fomin, Leningrad)
Consider equilateral triangle $ABC$ and suppose that there exist three distinct points $X, Y,Z$ lie inside triangle $ABC$ such that
i) $AX = BY = CZ$
ii) The triplets of points $(A,X,Z), (B,Y,X), (C,Z,Y )$ are collinear in that order.
Prove that $XY Z$ is an equilateral triangle.
On the horizontal line from left to right are the points $P, \, \, Q, \, \, R, \, \, S$. Construct a square $ABCD$, for which on the line $AD$ lies lies the point $P$, on the line $BC$ lies the point $Q$, on the line $AB$ lies the point $R$, on the line $CD$ lies the point $S $.
Triangle $A'B'C'$ is located inside triangle $ABC$ such that $AB \parallel A'B' $, $BC \parallel B'C'$ and $CA \parallel C'A'$ , and all three sides of these parallel sides are at distance $d$ at each case. Let $O$ and $O'$ be the centers of the inscribed circles of the triangles $ABC$ and $A'B'C'$ and $K$ and $K'$ are the the centers of their circumcircles. Prove that points $O, O', K$ and $K'$ lie on a straight line.
Let $P$ be an interior point of a circle $\Gamma$ centered at $O$ where $P \ne O$. Let $A$ and $B$ be distinct points on $\Gamma$. Lines $AP$ and $BP$ meet $\Gamma$ again at $C$ and $D$, respectively. Let $S$ be any interior point on line segment $PC$. The circumcircle of $\vartriangle ABS$ intersects line segment $PD$ at $T$. The line through $S$ perpendicular to $AC$ intersects $\Gamma$ at $U$ and $V$ . The line through $T$ perpendicular to $BD$ intersects $\Gamma$ at $X$ and $Y$ . Let $M$ and $N$ be the midpoints of $UV$ and $XY$ , respectively. Let $AM$ and $BN$ meet at $Q$. Suppose that $AB$ is not parallel to $CD$. Show that $P, Q$, and $O$ are collinear if and only if $S$ is the midpoint of $PC$.
Prove that if every line connecting any two points of some finite set of points of the plane contains at least one more point of this set, then all points of the set lie on one straight line.
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 $BC$.
Let $ABC$ be an acute, non-isosceles triangle with the circumcircle $(O)$. Denote $D, E$ as the midpoints of $AB,AC$ respectively. Two circles $(ABE)$ and $(ACD)$ intersect at $K$ differs from $A$. Suppose that the ray $AK$ intersects $(O)$ at $L$. The line $LB$ meets $(ABE)$ at the second point $M$ and the line $LC$ meets $(ACD)$ at the second point $N$.
a) Prove that $M, K, N$ collinear and $MN$ perpendicular to $OL$.
b) Prove that $K$ is the midpoint of $MN$
The inscribed circle $\Omega$ of triangle $ABC$ touches the sides $AB$ and $AC$ at points $K$ and $ L$, respectively. The line $BL$ intersects the circle $\Omega$ for the second time at the point $M$. The circle $\omega$ passes through the point $M$ and is tangent to the lines $AB$ and $BC$ at the points $P$ and $Q$, respectively. Let $N$ be the second intersection point of circles $\omega$ and $\Omega$, which is different from $M$. Prove that if $KM \parallel AC$ then the points $P, N$ and $L$ lie on one line.
Let $ABC$ be an acute triangle, and let $F_A$ and $F_B$ be the midpoints of sides $BC$ and $CA$, respectively. Let $E$ and $F$ be the intersection points of the circle centered at $F_A$ and passing through $A$ and the circle centered at $F_B$ and passing through $B$. Prove that if segments $CE$ and $CF$ have midpoints $N$ and $M$, respectively, then the intersection points of the circle centered at $M$ and passing through $E$ and the circle centered at $N$ and passing through $F$ lie on the line $AB$.
Given a triangle $ABC$ and points $P$ and $Q$. It is known that the triangles formed by the projections $P$ and $Q$ on the sides of $ABC$ are similar (vertices lying on the same sides of the original triangle correspond to each other). Prove that line $PQ$ passes through the center of the circumscribed circle of triangle $ABC$.
(A. Zaslavsky)
In the triangle $ ABC $, the bisectors of the internal and external angles are drawn at the vertices $ A $ and $ B $. Prove that the orthogonal projections of the point $ C $ on these bisectors lie on one straight line.