Found problems: 1065
2008 Oral Moscow Geometry Olympiad, 2
In a certain triangle, the bisectors of the two interior angles were extended to the intersection with the circumscribed circle and two equal chords were obtained. Is it true that the triangle is isosceles?
2001 Korea - Final Round, 2
Let $P$ be a given point inside a convex quadrilateral $O_1O_2O_3O_4$. For each $i = 1,2,3,4$, consider the lines $l$ that pass through $P$ and meet the rays $O_iO_{i-1}$ and $O_iO_{i+1}$ (where $O_0 = O_4$ and $O_5 = O_1$) at distinct points $A_i(l)$ and $B_i(l)$, respectively. Denote $f_i(l) = PA_i(l) \cdot PB_i(l)$. Among all such lines $l$, let $l_i$ be the one that minimizes $f_i$. Show that if $l_1 = l_3$ and $l_2 = l_4$, then the quadrilateral $O_1O_2O_3O_4$ is a parallelogram.
2013 Moldova Team Selection Test, 3
Consider the triangle $\triangle ABC$ with $AB \not = AC$. Let point $O$ be the circumcenter of $\triangle ABC$. Let the angle bisector of $\angle BAC$ intersect $BC$ at point $D$. Let $E$ be the reflection of point $D$ across the midpoint of the segment $BC$. The lines perpendicular to $BC$ in points $D,E$ intersect the lines $AO,AD$ at the points $X,Y$ respectively. Prove that the quadrilateral $B,X,C,Y$ is cyclic.
1990 All Soviet Union Mathematical Olympiad, 524
$A, B, C$ are adjacent vertices of a regular $2n$-gon and $D$ is the vertex opposite to $B$ (so that $BD$ passes through the center of the $2n$-gon). $X$ is a point on the side $AB$ and $Y$ is a point on the side $BC$ so that $XDY = \frac{\pi}{2n}$. Show that $DY$ bisects $\angle XYC$.
2006 ISI B.Stat Entrance Exam, 4
In the figure below, $E$ is the midpoint of the arc $ABEC$ and the segment $ED$ is perpendicular to the chord $BC$ at $D$. If the length of the chord $AB$ is $l_1$, and that of the segment $BD$ is $l_2$, determine the length of $DC$ in terms of $l_1, l_2$.
[asy]
unitsize(1 cm);
pair A=2dir(240),B=2dir(190),C=2dir(30),E=2dir(135),D=foot(E,B,C);
draw(circle((0,0),2)); draw(A--B--C); draw(E--D); draw(rightanglemark(C,D,E,8));
label("$A$",A,.5A); label("$B$",B,.5B); label("$C$",C,.5C); label("$E$",E,.5E); label("$D$",D,dir(-60));
[/asy]
1998 USAMTS Problems, 5
In the figure on the right, $O$ is the center of the circle, $OK$ and $OA$ are perpendicular to one another, $M$ is the midpoint of $OK$, $BN$ is parallel to $OK$, and $\angle AMN=\angle NMO$. Determine the measure of $\angle A B N$ in degrees.
[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair O=origin, A=dir(90), K=dir(180), M=0.5*dir(180), N=2/5*dir(90), B=dir(degrees((2/5, sqrt(21/25)))+90);
draw(K--O--A--M--N--B--A^^Circle(origin,1));
label("$A$", A, dir(O--A));
label("$K$", K, dir(O--K));
label("$B$", B, dir(O--B));
label("$N$", N, E);
label("$M$", M, S);
label("$O$", O, SE);[/asy]
2015 Caucasus Mathematical Olympiad, 3
Let $AL$ be the angle bisector of the acute-angled triangle $ABC$. and $\omega$ be the circle circumscribed about it. Denote by $P$ the intersection point of the extension of the altitude $BH$ of the triangle $ABC$ with the circle $\omega$ . Prove that if $\angle BLA= \angle BAC$, then $BP = CP$.
2021 All-Russian Olympiad, 4
In triangle $ABC$ angle bisectors $AA_{1}$ and $CC_{1}$ intersect at $I$. Line through $B$ parallel to $AC$ intersects rays $AA_{1}$ and $CC_{1}$ at points $A_{2}$ and $C_{2}$ respectively. Let $O_{a}$ and $O_{c}$ be the circumcenters of triangles $AC_{1}C_{2}$ and $CA_{1}A_{2}$ respectively. Prove that $\angle{O_{a}BO_{c}} = \angle{AIC} $
2002 AMC 12/AHSME, 23
In triangle $ ABC$, side $ AC$ and the perpendicular bisector of $ BC$ meet in point $ D$, and $ BD$ bisects $ \angle ABC$. If $ AD \equal{} 9$ and $ DC \equal{} 7$, what is the area of triangle $ ABD$?
$ \textbf{(A)}\ 14 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 14\sqrt5 \qquad \textbf{(E)}\ 28\sqrt5$
1987 All Soviet Union Mathematical Olympiad, 454
Vertex $B$ of the $\angle ABC$ lies out the circle, and the $[BA)$ and $[BC)$ beams intersect it. Point $K$ belongs to the intersection of the $[BA)$ beam and the circumference. Chord $KP$ is orthogonal to the angle bisector of $\angle ABC$ . Line $(KP)$ intersects the beam $BC$ in the point $M$. Prove that the segment $[PM]$ is twice as long as the distance from the circle centre to the angle bisector of $\angle ABC$ .
2009 Iran Team Selection Test, 5
$ ABC$ is a triangle and $ AA'$ , $ BB'$ and $ CC'$ are three altitudes of this triangle . Let $ P$ be the feet of perpendicular from $ C'$ to $ A'B'$ , and $ Q$ is a point on $ A'B'$ such that $ QA \equal{} QB$ . Prove that :
$ \angle PBQ \equal{} \angle PAQ \equal{} \angle PC'C$
2012 Sharygin Geometry Olympiad, 3
A circle with center $I$ touches sides $AB,BC,CA$ of triangle $ABC$ in points $C_{1},A_{1},B_{1}$. Lines $AI, CI, B_{1}I$ meet $A_{1}C_{1}$ in points $X, Y, Z$ respectively. Prove that $\angle Y B_{1}Z = \angle XB_{1}Z$.
2000 All-Russian Olympiad Regional Round, 10.7
In a convex quadrilateral $ABCD$ we draw the bisectors $\ell_a$, $\ell_b$, $\ell_c$, $\ell_d$ of external angles $A$, $B$, $C$, $D$ respectively. The intersection points of the lines $\ell_a$ and $\ell_b$, $\ell_b$ and $\ell_c$, $\ell_c$ and $\ell_d$, $\ell_d$ and $\ell_a$ are designated by $K$, $L$, $M$, $N$. It is known that $3$ perpendiculars drawn from $K$ on $AB$, from $L$ om $BC$, from $M$ on $CD$ intersect at one point. Prove that the quadrilateral $ABCD$ is cyclic.
2007 Korea - Final Round, 5
For the vertex $ A$ of a triangle $ ABC$, let $ l_a$ be the distance between the projections on $ AB$ and $ AC$ of the intersection of the angle bisector of ∠$ A$ with side $ BC$. Define $ l_b$ and $ l_c$ analogously. If $ l$ is the perimeter of triangle $ ABC$, prove that $ \frac{l_a l_b l_c}{l^3}\le\frac{1}{64}$.
2023 Macedonian Balkan MO TST, Problem 3
Let $ABC$ be a triangle such that $AB<AC$. Let $D$ be a point on the segment $BC$ such that $BD<CD$. The angle bisectors of $\angle ADB$ and $\angle ADC$ meet the segments $AB$ and $AC$ at $E$ and $F$ respectively. Let $\omega$ be the circumcircle of $AEF$ and $M$ be the midpoint of $EF$. The ray $AD$ meets $\omega$ at $X$ and the line through $X$ parallel to $EF$ meets $\omega$ again at $Y$. If $YM$ meets $\omega$ at $T$, show that $AT$, $EF$ and $BC$ are concurrent.
[i]Authored by Nikola Velov[/i]
2008 Indonesia TST, 3
Let $\Gamma_1$ and $\Gamma_2$ be two circles that tangents each other at point $N$, with $\Gamma_2$ located inside $\Gamma_1$. Let $A, B, C$ be distinct points on $\Gamma_1$ such that $AB$ and $AC$ tangents $\Gamma_2$ at $D$ and $E$, respectively. Line $ND$ cuts $\Gamma_1$ again at $K$, and line $CK$ intersects line $DE$ at $I$.
(i) Prove that $CK$ is the angle bisector of $\angle ACB$.
(ii) Prove that $IECN$ and $IBDN$ are cyclic quadrilaterals.
2021 JHMT HS, 6
Suppose $JHMT$ is a convex quadrilateral with perimeter $68$ and satisfies $\angle HJT = 120^\circ,$ $HM = 20,$ and $JH + JT = JM > HM.$ Furthermore, $\overrightarrow{JM}$ bisects $\angle HJT.$ Compute $JM.$
2008 Costa Rica - Final Round, 2
Let $ ABC$ be a triangle and let $ P$ be a point on the angle bisector $ AD$, with $ D$ on $ BC$. Let $ E$, $ F$ and $ G$ be the intersections of $ AP$, $ BP$ and $ CP$ with the circumcircle of the triangle, respectively. Let $ H$ be the intersection of $ EF$ and $ AC$, and let $ I$ be the intersection of $ EG$ and $ AB$. Determine the geometric place of the intersection of $ BH$ and $ CI$ when $ P$ varies.
2004 Bosnia and Herzegovina Junior BMO TST, 5
In the isosceles triangle $ABC$ ($AC = BC$), $AB =\sqrt3$ and the altitude $CD =\sqrt2$. Let $E$ and $F$ be the midpoints of $BC$ and $DB$, respectively and $G$ be the intersection of $AE$ and $CF$. Prove that $D$ belongs to the angle bisector of $\angle AGF$.
2022 Yasinsky Geometry Olympiad, 4
Let $X$ be an arbitrary point on side $BC$ of triangle $ABC$. Triangle $T$ is formed by the angle bisectors of the angles $\angle ABC$, $\angle ACB$ and $\angle AXC$. Prove that the circle circumscribed around the triangle $T$, passes through the vertex $A$.
(Dmytro Prokopenko)
2010 AIME Problems, 15
In triangle $ ABC$, $ AC \equal{} 13, BC \equal{} 14,$ and $ AB\equal{}15$. Points $ M$ and $ D$ lie on $ AC$ with $ AM\equal{}MC$ and $ \angle ABD \equal{} \angle DBC$. Points $ N$ and $ E$ lie on $ AB$ with $ AN\equal{}NB$ and $ \angle ACE \equal{} \angle ECB$. Let $ P$ be the point, other than $ A$, of intersection of the circumcircles of $ \triangle AMN$ and $ \triangle ADE$. Ray $ AP$ meets $ BC$ at $ Q$. The ratio $ \frac{BQ}{CQ}$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\minus{}n$.
2006 Tournament of Towns, 4
Given triangle $ABC, BC$ is extended beyond $B$ to the point $D$ such that $BD = BA$. The bisectors of the exterior angles at vertices $B$ and $C$ intersect at the point $M$. Prove that quadrilateral $ADMC$ is cyclic. (4)
Geometry Mathley 2011-12, 14.2
The nine-point Euler circle of triangle $ABC$ is tangent to the excircles in the angle $A,B,C$ at $Fa, Fb, Fc$ respectively. Prove that $AF_a$ bisects the angle $\angle CAB$ if and only if $AFa$ bisects the angle $\angle F_bAF_c$.
Đỗ Thanh Sơn
Denmark (Mohr) - geometry, 2013.5
The angle bisector of $A$ in triangle $ABC$ intersects $BC$ in the point $D$. The point $E$ lies on the side $AC$, and the lines $AD$ and $BE$ intersect in the point $F$. Furthermore, $\frac{|AF|}{|F D|}= 3$ and $\frac{|BF|}{|F E|}=\frac{5}{3}$. Prove that $|AB| = |AC|$.
[img]https://1.bp.blogspot.com/-evofDCeJWPY/XzT9dmxXzVI/AAAAAAAAMVY/ZN87X3Cg8iMiULwvMhgFrXbdd_f1f-JWwCLcBGAsYHQ/s0/2013%2BMohr%2Bp5.png[/img]
1998 Baltic Way, 14
Given triangle $ABC$ with $AB<AC$. The line passing through $B$ and parallel to $AC$ meets the external angle bisector of $\angle BAC$ at $D$. The line passing through $C$ and parallel to $AB$ meets this bisector at $E$. Point $F$ lies on the side $AC$ and satisfies the equality $FC=AB$. Prove that $DF=FE$.