Found problems: 1065
2020 Dutch IMO TST, 2
Given is a triangle $ABC$ with its circumscribed circle and $| AC | <| AB |$. On the short arc $AC$, there is a variable point $D\ne A$. Let $E$ be the reflection of $A$ wrt the inner bisector of $\angle BDC$. Prove that the line $DE$ passes through a fixed point, regardless of point $D$.
Ukraine Correspondence MO - geometry, 2015.8
On the sides $BC, AC$ and $AB$ of the equilateral triangle $ABC$ mark the points $D, E$ and $F$ so that $\angle AEF = \angle FDB$ and $\angle AFE = \angle EDC$. Prove that $DA$ is the bisector of the angle $EDF$.
2023 4th Memorial "Aleksandar Blazhevski-Cane", P3
Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\omega$ with center $O$. The lines $AD$ and $BC$ meet at $E$, while the lines $AB$ and $CD$ meet at $F$. Let $P$ be a point on the segment $EF$ such that $OP \perp EF$. The circle $\Gamma_{1}$ passes through $A$ and $E$ and is tangent to $\omega$ at $A$, while $\Gamma_{2}$ passes through $C$ and $F$ and is tangent to $\omega$ at $C$. If $\Gamma_{1}$ and $\Gamma_{2}$ meet at $X$ and $Y$, prove that $PO$ is the bisector of $\angle XPY$.
[i]Proposed by Nikola Velov[/i]
1987 IMO Longlists, 40
The perpendicular line issued from the center of the circumcircle to the bisector of angle $C$ in a triangle $ABC$ divides the segment of the bisector inside $ABC$ into two segments with ratio of lengths $\lambda$. Given $b = AC$ and $a = BC$, find the length of side $c.$
2021 Turkey MO (2nd round), 4
Points $D$ and $E$ are taken on $[BC]$ and $[AC]$ of acute angled triangle $ABC$ such that $BD$ and $CE$ are angle bisectors. Projections of $D$ onto $BC$ and $BA$ are $P$ and $Q$, projections of $E$ onto $CA$ and $CB$ are $R$ and $S$. Let $AP \cap CQ=X$, $AS \cap BR=Y$ and $BX \cap CY=Z$. Show that $AZ \perp BC$.
2000 National Olympiad First Round, 33
Let $K$ be a point on the side $[AB]$, and $L$ be a point on the side $[BC]$ of the square $ABCD$. If $|AK|=3$, $|KB|=2$, and the distance of $K$ to the line $DL$ is $3$, what is $|BL|:|LC|$?
$ \textbf{(A)}\ \frac78
\qquad\textbf{(B)}\ \frac{\sqrt 3}2
\qquad\textbf{(C)}\ \frac 87
\qquad\textbf{(D)}\ \frac 38
\qquad\textbf{(E)}\ \frac{\sqrt 2}2
$
2016 Dutch IMO TST, 3
Let $\vartriangle ABC$ be an isosceles triangle with $|AB| = |AC|$. Let $D, E$ and $F$ be points on line segments $BC, CA$ and $AB$, respectively, such that $|BF| = |BE|$ and such that $ED$ is the internal angle bisector of $\angle BEC$. Prove that $|BD|= |EF|$ if and only if $|AF| = |EC|$.
2005 All-Russian Olympiad Regional Round, 8.6
In quadrilateral $ABCD$, angles $A$ and $C$ are equal. Angle bisector of $B$ intersects line $AD$ at point $P$. Perpendicular on $BP$ passing through point $A$ intersects line $BC$ at point $Q$. Prove that the lines $PQ$ and $CD$ are parallel.
2009 Sharygin Geometry Olympiad, 3
The cirumradius and the inradius of triangle $ABC$ are equal to $R$ and $r, O, I$ are the centers of respective circles. External bisector of angle $C$ intersect $AB$ in point $P$. Point $Q$ is the projection of $P$ to line $OI$. Find distance $OQ.$
(A.Zaslavsky, A.Akopjan)
2019 Hanoi Open Mathematics Competitions, 5
Let $ABC$ be a triangle and $AD$ be the bisector of the triangle ($D \in (BC)$) Assume that $AB =14$ cm,
$AC = 35$ cm and $AD = 12$ cm; which of the following is the area of triangle $ABC$ in cm$^2$?
[b]A.[/b] $\frac{1176}{5}$ [b]B.[/b] $\frac{1167}{5}$ [b]C.[/b] $234$ [b]D.[/b] $\frac{1176}{7}$ [b]E.[/b] $236$
2008 Bundeswettbewerb Mathematik, 3
Prove: In an acute triangle $ ABC$ angle bisector $ w_{\alpha},$ median $ s_b$ and the altitude $ h_c$ intersect in one point if $ w_{\alpha},$ side $ BC$ and the circle around foot of the altitude $ h_c$ have vertex $ A$ as a common point.
1991 USAMO, 1
In triangle $\, ABC, \,$ angle $\,A\,$ is twice angle $\,B,\,$ angle $\,C\,$ is obtuse, and the three side lengths $\,a,b,c\,$ are integers. Determine, with proof, the minimum possible perimeter.
2020 Ukrainian Geometry Olympiad - April, 4
Inside triangle $ABC$, the point $P$ is chosen such that $\angle PAB = \angle PCB =\frac14 (\angle A+ \angle C)$. Let $BL$ be the bisector of $\vartriangle ABC$. Line $PL$ intersects the circumcircle of $\vartriangle APC$ at point $Q$. Prove that the line $QB$ is the bisector of $\angle AQC$.
2020 Macedonian NationŠ°l Olympiad, 3
Let $ABC$ be a triangle, and $A_1, B_1, C_1$ be points on the sides $BC, CA, AB$, respectively, such that $AA_1, BB_1, CC_1$ are the internal angle bisectors of $\triangle ABC$. The circumcircle $k' = (A_1B_1C_1)$ touches the side $BC$ at $A_1$. Let $B_2$ and $C_2$, respectively, be the second intersection points of $k'$ with lines $AC$ and $AB$. Prove that $|AB| = |AC|$ or $|AC_1| = |AB_2|$.
2016 SDMO (High School), 4
Let triangle $ABC$ be an isosceles triangle with $AB = AC$. Suppose that the angle bisector of its angle $\angle B$ meets the side $AC$ at a point $D$ and that $BC = BD+AD$.
Determine $\angle A$.
Denmark (Mohr) - geometry, 2018.5
In triangle $ABC$ the angular bisector from $A$ intersects the side $BC$ at the point $D$, and the angular bisector from $B$ intersects the side $AC$ at the point $E$. Furthermore $|AE| + |BD| = |AB|$. Prove that $\angle C = 60^o$
[img]https://1.bp.blogspot.com/-8ARqn8mLn24/XzP3P5319TI/AAAAAAAAMUQ/t71-imNuS18CSxTTLzYXpd806BlG5hXxACLcBGAsYHQ/s0/2018%2BMohr%2Bp5.png[/img]
JBMO Geometry Collection, 2001
Let $ABC$ be a triangle with $\angle C = 90^\circ$ and $CA \neq CB$. Let $CH$ be an altitude and $CL$ be an interior angle bisector. Show that for $X \neq C$ on the line $CL$, we have $\angle XAC \neq \angle XBC$. Also show that for $Y \neq C$ on the line $CH$ we have $\angle YAC \neq \angle YBC$.
[i]Bulgaria[/i]
2013 NIMO Problems, 12
In $\triangle ABC$, $AB = 40$, $BC = 60$, and $CA = 50$. The angle bisector of $\angle A$ intersects the circumcircle of $\triangle ABC$ at $A$ and $P$. Find $BP$.
[i]Proposed by Eugene Chen[/i]
1998 Italy TST, 2
In a triangle $ABC$, points $H,M,L$ are the feet of the altitude from $C$, the median from $A$, and the angle bisector from $B$, respectively. Show that if triangle $HML$ is equilateral, then so is triangle $ABC$.
2015 Tuymaada Olympiad, 2
$D$ is midpoint of $AC$ for $\triangle ABC$. Bisectors of $\angle ACB,\angle ABD$ are perpendicular. Find max value for $\angle BAC$
[i](S. Berlov)[/i]
2023 Dutch Mathematical Olympiad, 4
In acute-angled triangle $ABC$ with $|BC| < |BA|$, point $N$ is the midpoint of $AC$. The circle with diameter $AB$ intersects the bisector of $\angle B$ in two points: $B$ and $X$. Prove that $XN$ is parallel to $BC$.
[img]https://cdn.artofproblemsolving.com/attachments/5/1/f0ae8f5df8f2cc1bb80de1ee1807dc845a87b3.png[/img]
2014 China National Olympiad, 1
Let $ABC$ be a triangle with $AB>AC$. Let $D$ be the foot of the internal angle bisector of $A$. Points $F$ and $E$ are on $AC,AB$ respectively such that $B,C,F,E$ are concyclic. Prove that the circumcentre of $DEF$ is the incentre of $ABC$ if and only if $BE+CF=BC$.
1999 Belarusian National Olympiad, 7
Let [i]O[/i] be the center of circle[i] W[/i]. Two equal chords [i]AB[/i] and [i]CD [/i]of[i] W [/i]intersect at [i]L [/i]such that [i]AL>LB [/i]and [i]DL>LC[/i]. Let [i]M [/i]and[i] N [/i]be points on [i]AL[/i] and [i]DL[/i] respectively such that ([i]ALC[/i])=2*([i]MON[/i]). Prove that the chord of [i]W[/i] passing through [i]M [/i]and [i]N[/i] is equal to [i]AB[/i] and [i]CD[/i].
2012 Tournament of Towns, 5
Let $\ell$ be a tangent to the incircle of triangle $ABC$. Let $\ell_a,\ell_b$ and $\ell_c$ be the respective images of $\ell$ under reflection across the exterior bisector of $\angle A,\angle B$ and $\angle C$. Prove that the triangle formed by these lines is congruent to $ABC$.
2011 Saudi Arabia Pre-TST, 1.3
The quadrilateral $ABCD$ has $AD = DC = CB < AB$ and $AB \parallel CD$. Points $E$ and $F$ lie on the sides $CD$ and $BC$ such that $\angle ADE = \angle AEF$. Prove that:
(a) $4CF \le CB$.
(b) If $4CF = CB$, then $AE$ is the angle bisector of $\angle DAF$.