Found problems: 250
Let $\Gamma$ a circumference. $T$ a point in $\Gamma$, $P$ and $A$ two points outside $\Gamma$ such that $PT$ is tangent to $\Gamma$ and $PA=PT$. Let $C$ a point in $\Gamma (C\neq T)$, $AC$ and $PC$ intersect again $\Gamma$ in $D$ and $B$, respectively. $AB$ intersect $\Gamma$ in $E$. Prove that $DE$ it´s parallel to $AP$
Let $\Delta ABC$ be a scalene triangle. Points $D,E$ lie on side $\overline{AC}$ in the order, $A,E,D,C$. Let the parallel through $E$ to $BC$ intersect $\odot (ABD)$ at $F$, such that, $E$ and $F$ lie on the same side of $AB$. Let the parallel through $E$ to $AB$ intersect $\odot (BDC)$ at $G$, such that, $E$ and $G$ lie on the same side of $BC$. Prove, Points $D,F,E,G$ are concyclic
Let $H$ be the orthocenter of the acute triangle $ABC$. Let $H_a$ be the foot of the perpendicular from $A$ to $BC$ and let the line through $H$ parallel to $BC$ intersect the circle with diameter $AH_a$ in the points $P_a$ and $Q_a$. Similarly, we define the points $P_b, Q_b$ and $P_c,Q_c$.
Show that the six points $P_a,Q_a,P_b,Q_b,P_c,Q_c$ lie on a common circle.
Let \(ABC\) be an isosceles triangle with \(AB = BC\). Let \(D\) be a point on segment \(AB\), \(E\) be a point on segment \(BC\), and \(P\) be a point on segment \(DE\) such that \(AD = DP\) and \(CE = PE\). Let \(M\) be the midpoint of \(DE\). The line parallel to \(AB\) through \(M\) intersects \(AC\) at \(X\) and the line parallel to \(BC\) through \(M\) intersects \(AC\) at \(Y\). The lines \(DX\) and \(EY\) intersect at \(F\). Prove that \(FP\) is perpendicular to \(DE\).
Given a square $ABCD$, points $E$ and $F$ are taken inside the segments $BC$ and $CD$ so that $\angle EAF = 45^o$. The lines $AE$ and $AF$ intersect the circle circumscribed to the square at points $G$ and $H$ respectively. Prove that lines $EF$ and $GH$ are parallel.
Let $n$ and $c$ be coprime positive integers. For any integer $i$, denote by $i' $ the remainder of division of product $ci$ by $n$. Let $A_o.A_1,A_2,...,A_{n-1}$ be a regular $n$-gon. Prove that
a) if $A_iA_j \parallel A_kA_i$ then $A_{i'}A_{j'} \parallel A_{k'}A_{i'}$
b) if $A_iA_j \perp A_kA_l$ then $A_{i'}A_{j'} \perp A_{k'}A_{l'}$
$P_1$, $P_2$, $P_3$, $Q_1$, $Q_2$, $Q_3$ are distinct points in the plane. The distances $P_1Q_1$, $P_2Q_2$, $P_3Q_3$ are equal. $P_1P_2$ and $Q_2Q_1$ are parallel (not antiparallel), similarly $P_1P_3$ and $Q_3Q_1$, and $P_2P_3$ and $Q_3Q_2$. Show that $P_1Q_1$, $P_2Q_2$ and $P_3Q_3$ intersect in a point.
The line $\ell$ parallel to the side $BC$ of the triangle $ABC$, intersects its sides $AB,AC$ at the points $D,E$, respectively. The circumscribed circle of triangle $ABC$ intersects line $\ell$ at points $F$ and $G$, such that points $F,D,E,G$ lie on line $\ell$ in this order. The circumscribed circles of the triangles $FEB$ and $DGC$ intersect at points $P$ and $Q$. Prove that points $A, P$ and $Q$ are collinear.
Let $ABC$ be a triangle and $D$ a point on the side $BC$. The angle bisectors of $\angle ADB ,\angle ADC$ intersect $AB ,AC$ at points $M ,N$ respectively. The angle bisectors of $\angle ABD , \angle ACD$ intersects $DM , DN$ at points $K , L$ respectively. Prove that $AM = AN$ if and only if $MN$ and $KL$ are parallel.
Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.
The excircle of right-angled triangle $ABC$ ($\angle B =90^o$) touches side $BC$ at point $A_1$ and touches line $AC$ in point $A_2$. Line $A_1A_2$ meets the incircle of $ABC$ for the first time at point $A'$, point $C'$ is defined similarly. Prove that $AC||A'C'$.
In an isosceles triangle $ABC$ in which $AC = BC$ and $\angle ABC < 60^o$, $I$ and $O$ are the centers of the inscribed and circumscribed circles, respectively. For the triangle $BIO$, the circumscribed circle intersects the side $BC$ again at $D$. Prove that:
i) lines $AC$ and $DI$ are parallel,
ii) lines $OD$ and $IB$ are perpendicular.
Consider two equilateral triangles $ABC$ and $MNP$ with the property that $AB \parallel MN, BC \parallel NP$ and $CA \parallel PM$ , so that the surfaces of the triangles intersect after a convex hexagon. The distances between the three pairs of parallel lines are at most equal to $1$. Show that at least one of the two triangles has the side at most equal to $\sqrt {3}$ .
Let $AD, BE, CF$ be segments whose midpoints are on the same line $\ell$. The points $X, Y, Z$ lie on the lines $EF, FD, DE$ respectively such that $AX \parallel BY \parallel CZ \parallel \ell$. Prove that $X, Y, Z$ are collinear.
Tran Quang Hung, High School of Natural Sciences, Hanoi National University
Let $BH_b, CH_c$ be altitudes of an acute-angled triangle $ABC$. The line $H_bH_c$ meets the circumcircle of $ABC$ at points $X$ and $Y$. Points $P,Q$ are the reflections of $X,Y$ about $AB,AC$ respectively. Prove that $PQ \parallel BC$.
[i]Proposed by Pavel Kozhevnikov[/i]
Let $ABC$ be a triangle with centroid $G$, and let $E$ and $F$ be points on side $BC$ such that $BE = EF = F C$. Points $X$ and $Y$ lie on lines $AB$ and $AC$, respectively, so that $X$, $Y$ , and $G$ are not collinear. If the line through $E$ parallel to $XG$ and the line through $F$ parallel to $Y G$ intersect at $P\ne G$, prove that $GP$ passes through the midpoint of $XY$.
Each diagonal of a convex pentagon is parallel to one of its sides. Prove that the ratio of the length of each diagonal to the length of the corresponding parallel side is the same, and find this ratio.
It is given triangle $ABC$ such that $\angle ABC = 3 \angle CAB$. On side $AC$ there are two points $M$ and $N$ in order $A - N - M - C$ and $\angle CBM = \angle MBN = \angle NBA$. Let $L$ be an arbitrary point on side $BN$ and $K$ point on $BM$ such that $LK \mid \mid AC$. Prove that lines $AL$, $NK$ and $BC$ are concurrent
Let $ABCDEF$ be a hexagon inscribed in a circle (with vertices in that order) with $\angle B + \angle C > 180^o$ and $\angle E + \angle F > 180^o$. Let the lines $AB$ and $CD$ intersect at $X$ and the lines $AF$ and $DE$ intersect at $S$. Let $XY$ and $ST$ be the diameters of the circumcircles of $\vartriangle BCX$ and $\vartriangle EFS$ respectively. If $U$ is the intersection point of the lines $BX$ and $ES$ and $V$ is the intersection point of the lines $BY$ and $ET,$ prove that the lines $UV, XY$ and $ST$ are all parallel.
Let $ABC$ be an acute triangle with circumcenter $O$, and let $D$, $E$, and $F$ be the feet of altitudes from $A$, $B$, and $C$ to sides $BC$, $CA$, and $AB$, respectively. Denote by $P$ the intersection of the tangents to the circumcircle of $ABC$ at $B$ and $C$. The line through $P$ perpendicular to $EF$ meets $AD$ at $Q$, and let $R$ be the foot of the perpendicular from $A$ to $EF$. Prove that $DR$ and $OQ$ are parallel.
Let \( O \) and \( H \) be the circumcenter and orthocenter of the acute triangle \( ABC \). On sides \( AC \) and \( AB \), points \( D \) and \( E \) are chosen respectively such that segment \( DE \) passes through point \( O \) and \( DE \parallel BC \). On side \( BC \), points \( X \) and \( Y \) are chosen such that \( BX = OD \) and \( CY = OE \). Prove that \( \angle XHY + 2\angle BAC = 180^\circ \).
[i]Proposed by Matthew Kurskyi[/i]
Let $ABC$ be an acute, non-isosceles triangle with $H$ the orthocenter and $M$ the midpoint of $AH$. Denote $O_1$,$O_2$ as the centers of circles pass through $H$ and respectively tangent to $BC$ at $B$, $C$. Let $X$, $Y$ be the ex-centers which respect to angle $H$ in triangles $HMO_1$,$HMO_2$. Prove that $XY$ is parallel to $O_1O_2$.
Given an acute-angled triangle $ABC$. A straight line parallel to $BC$ intersects sides $AB$ and $AC$ at points $M$ and $P$, respectively. At what location of the points $M$ and $P$ will the radius of the circle circumscribed about the triangle $BMP$ be the smallest?
(I. Sharygin)
In quadrilateral $ABCD$ sides $BC$ and $AD$ are parallel. In each of the four vertices we draw an angular bisector. The angular bisectors of angles $A$ and $B$ intersect in point $P$, those of angles $B$ and $C$ intersect in point $Q$, those of angles $C$ and $D$ intersect in point $R$, and those of angles $D$ and $A$ intersect in point S. Suppose that $PS$ is parallel to $QR$. Prove that $|AB| =|CD|$.
[asy]
unitsize(1.2 cm);
pair A, B, C, D, P, Q, R, S;
A = (0,0);
D = (3,0);
B = (0.8,1.5);
C = (3.2,1.5);
S = extension(A, incenter(A,B,D), D, incenter(A,C,D));
Q = extension(B, incenter(A,B,C), C, C + incenter(A,B,D) - A);
P = extension(A, S, B, Q);
R = extension(D, S, C, Q);
draw(A--D--C--B--cycle);
draw(B--Q--C);
draw(A--S--D);
dot("$A$", A, SW);
dot("$B$", B, NW);
dot("$C$", C, NE);
dot("$D$", D, SE);
dot("$P$", P, dir(90));
dot("$Q$", Q, dir(270));
dot("$R$", R, dir(90));
dot("$S$", S, dir(90));
[/asy]
Attention: the figure is not drawn to scale.
Let $ABC$ be a triangle with $\angle ACB=90^{\circ}$. Suppose that the tangent line at $C$ to the circle passing through $A,B,C$ intersects the line $AB$ at $D$. Let $E$ be the midpoint of $CD$ and let $F$ be a point on $EB$ such that $AF$ is parallel to $CD$.
Prove that the lines $AB$ and $CF$ are perpendicular.