Found problems: 1065
2017 Junior Balkan Team Selection Tests - Romania, 3
Let $I$ be the incenter of the scalene $\Delta ABC$, such, $AB<AC$, and let $I'$ be the reflection of point $I$ in line $BC$. The angle bisector $AI$ meets $BC$ at $D$ and circumcircle of $\Delta ABC$ at $E$. The line $EI'$ meets the circumcircle at $F$. Prove, that,
$\text{(i) } \frac{AI}{IE}=\frac{ID}{DE}$
$\text{(ii) } IA=IF$
2021 Korea Junior Math Olympiad, 4
In an acute triangle $ABC$ with $\overline{AB} < \overline{AC}$, angle bisector of $A$ and perpendicular bisector of $\overline{BC}$ intersect at $D$. Let $P$ be an interior point of triangle $ABC$. Line $CP$ meets the circumcircle of triangle $ABP$ again at $K$. Prove that $B, D, K$ are collinear if and only if $AD$ and $BC$ meet on the circumcircle of triangle $APC$.
1967 IMO Longlists, 9
Circle $k$ and its diameter $AB$ are given. Find the locus of the centers of circles inscribed in the triangles having one vertex on $AB$ and two other vertices on $k.$
1999 Mexico National Olympiad, 5
In a quadrilateral $ABCD$ with $AB // CD$, the external bisectors of the angles at $B$ and $C$ meet at $P$, while the external bisectors of the angles at $A$ and $D$ meet at $Q$. Prove that the length of $PQ$ equals the semiperimeter of $ABCD$.
2012 Oral Moscow Geometry Olympiad, 6
Restore the triangle with a compass and a ruler given the intersection point of altitudes and the feet of the median and angle bisectors drawn to one side. (No research required.)
2021 Iran Team Selection Test, 5
Point $X$ is chosen inside the non trapezoid quadrilateral $ABCD$ such that $\angle AXD +\angle BXC=180$.
Suppose the angle bisector of $\angle ABX$ meets the $D$-altitude of triangle $ADX$ in $K$, and the angle bisector of $\angle DCX$ meets the $A$-altitude of triangle $ADX$ in $L$.We know $BK \perp CX$ and $CL \perp BX$. If the circumcenter of $ADX$ is on the line $KL$ prove that $KL \perp AD$.
Proposed by [i]Alireza Dadgarnia[/i]
1953 AMC 12/AHSME, 28
In triangle $ ABC$, sides $ a,b$ and $ c$ are opposite angles $ A,B$ and $ C$ respectively. $ AD$ bisects angle $ A$ and meets $ BC$ at $ D$. Then if $ x \equal{} \overline{CD}$ and $ y \equal{} \overline{BD}$ the correct proportion is:
$ \textbf{(A)}\ \frac {x}{a} \equal{} \frac {a}{b \plus{} c} \qquad\textbf{(B)}\ \frac {x}{b} \equal{} \frac {a}{a \plus{} c} \qquad\textbf{(C)}\ \frac {y}{c} \equal{} \frac {c}{b \plus{} c} \\
\textbf{(D)}\ \frac {y}{c} \equal{} \frac {a}{b \plus{} c} \qquad\textbf{(E)}\ \frac {x}{y} \equal{} \frac {c}{b}$
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.
2002 Turkey Team Selection Test, 2
In a triangle $ABC$, the angle bisector of $\widehat{ABC}$ meets $[AC]$ at $D$, and the angle bisector of $\widehat{BCA}$ meets $[AB]$ at $E$. Let $X$ be the intersection of the lines $BD$ and $CE$ where $|BX|=\sqrt 3|XD|$ ve $|XE|=(\sqrt 3 - 1)|XC|$. Find the angles of triangle $ABC$.
1996 USAMO, 3
Let $ABC$ be a triangle. Prove that there is a line $\ell$ (in the plane of triangle $ABC$) such that the intersection of the interior of triangle $ABC$ and the interior of its reflection $A'B'C'$ in $\ell$ has area more than $\frac23$ the area of triangle $ABC$.
1999 Tournament Of Towns, 5
The sides $AB$ and $AC$ are tangent at points $P$ and $Q$, respectively, to the incircle of a triangle $ABC. R$ and $S$ are the midpoints of the sides $AC$ and $BC$, respectively, and $T$ is the intersection point of the lines $PQ$ and $RS$. Prove that $T$ lies on the bisector of the angle $B$ of the triangle.
(M Evdokimov)
1998 AIME Problems, 12
Let $ABC$ be equilateral, and $D, E,$ and $F$ be the midpoints of $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively. There exist points $P, Q,$ and $R$ on $\overline{DE}, \overline{EF},$ and $\overline{FD},$ respectively, with the property that $P$ is on $\overline{CQ}, Q$ is on $\overline{AR},$ and $R$ is on $\overline{BP}.$ The ratio of the area of triangle $ABC$ to the area of triangle $PQR$ is $a+b\sqrt{c},$ where $a, b$ and $c$ are integers, and $c$ is not divisible by the square of any prime. What is $a^{2}+b^{2}+c^{2}$?
Estonia Open Junior - geometry, 2006.1.3
Let ABCD be a parallelogram, M the midpoint of AB and N the intersection of CD
and the angle bisector of ABC. Prove that CM and BN are perpendicular iff AN is the angle bisector of DAB.
2007 Hanoi Open Mathematics Competitions, 6
In triangle $ABC, \angle BAC = 60^o, \angle ACB = 90^o$ and $D$ is on $BC$.
If $AD$ bisects $\angle BAC$ and $CD = 3$ cm, calculate $DB$ .
2018 Bulgaria JBMO TST, 2
Let $ABC$ be a triangle and $AA_1$ be the angle bisector of $A$ ($A_1 \in BC$). The point $P$ is on the segment $AA_1$ and $M$ is the midpoint of the side $BC$. The point $Q$ is on the line connecting $P$ and $M$ such that $M$ is the midpoint of $PQ$. Define $D$ and $E$ as the intersections of $BQ$, $AC$, and $CQ$, $AB$. Prove that $CD=BE$.
Kyiv City MO Juniors Round2 2010+ geometry, 2016.9.2
The bisector of the angle $BAC$of the acute triangle $ABC$ ( $AC \ne AB$) intersects its circumscribed circle for the second time at the point $W$. Let $O$ be the center of the circumscribed circle $\Delta ABC$. The line $AW$ intersects for the second time the circumcribed circles of triangles $OWB$ and $OWC$ at the points $N$ and $M$, respectively. Prove that $BN + MC = AW$.
(Mitrofanov V., Hilko D.)
2011 Dutch BxMO TST, 2
In an acute triangle $ABC$ the angle $\angle C$ is greater than $\angle A$. Let $E$ be such that $AE$ is a diameter of the circumscribed circle $\Gamma$ of \vartriangle ABC. Let $K$ be the intersection of $AC$ and the tangent line at $B$ to $\Gamma$. Let $L$ be the orthogonal projection of $K$ on $AE$ and let $D$ be the intersection of $KL$ and $AB$. Prove that $CE$ is the bisector of $\angle BCD$.
2001 Hungary-Israel Binational, 5
In a triangle $ABC$ , $B_{1}$ and $C_{1}$ are the midpoints of $AC$ and $AB$ respectively, and $I$ is the incenter. The lines $B_{1}I$ and $C_{1}I$ meet $AB$ and $AC$ respectively at $C_{2}$ and $B_{2}$ . If the areas of $\Delta ABC$ and $\Delta AB_{2}C_{2}$ are equal, find $\angle{BAC}$ .
2019 New Zealand MO, 2
Let $X$ be the intersection of the diagonals $AC$ and $BD$ of convex quadrilateral $ABCD$. Let $P$ be the intersection of lines $AB$ and $CD$, and let $Q$ be the intersection of lines $PX$ and $AD$. Suppose that $\angle ABX = \angle XCD = 90^o$. Prove that $QP$ is the angle bisector of $\angle BQC$.
2023 Korea Summer Program Practice Test, P6
$AB < AC$ on $\triangle ABC$. The midpoint of arc $BC$ which doesn't include $A$ is $T$ and which includes $A$ is $S$. On segment $AB,AC$, $D,E$ exist so that $DE$ and $BC$ are parallel. The outer angle bisector of $\angle ABE$ and $\angle ACD$ meets $AS$ at $P$ and $Q$. Prove that the circumcircle of $\triangle PBE$ and $\triangle QCD$ meets on $AT$.
1968 German National Olympiad, 6
Prove the following two statements:
(a) If a triangle is isosceles, then two of its bisectors are of equal length.
(b) If two angle bisectors in a triangle are of equal length, then it is isosceles.
2017 239 Open Mathematical Olympiad, 6
Given a circumscribed quadrilateral $ABCD$ in which $$\sqrt{2}(BC-BA)=AC.$$ Let $X$ be the midpoint of $AC$ and $Y$ a point on the angle bisector of $B$ such that $XD$ is the angle bisector of $BXY$. Prove that $BD$ is tangent to the circumcircle of $DXY$.
2010 ELMO Shortlist, 1
Let $ABC$ be a triangle. Let $A_1$, $A_2$ be points on $AB$ and $AC$ respectively such that $A_1A_2 \parallel BC$ and the circumcircle of $\triangle AA_1A_2$ is tangent to $BC$ at $A_3$. Define $B_3$, $C_3$ similarly. Prove that $AA_3$, $BB_3$, and $CC_3$ are concurrent.
[i]Carl Lian.[/i]
2004 National Olympiad First Round, 25
Let $D$ be the foot of the internal angle bisector of the angle $A$ of a triangle $ABC$. Let $E$ be a point on side $[AC]$ such that $|CE|= |CD|$ and $|AE|=6\sqrt 5$; let $F$ be a point on the ray $[AB$ such that $|DB|=|BF|$ and $|AB|<|AF| = 8\sqrt 5$. What is $|AD|$?
$
\textbf{(A)}\ 10\sqrt 5
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 4\sqrt{15}
\qquad\textbf{(D)}\ 7\sqrt 5
\qquad\textbf{(E)}\ \text{None of above}
$
2009 China Team Selection Test, 2
In convex quadrilateral $ ABCD$, $ CB,DA$ are external angle bisectors of $ \angle DCA,\angle CDB$, respectively. Points $ E,F$ lie on the rays $ AC,BD$ respectively such that $ CEFD$ is cyclic quadrilateral. Point $ P$ lie in the plane of quadrilateral $ ABCD$ such that $ DA,CB$ are external angle bisectors of $ \angle PDE,\angle PCF$ respectively. $ AD$ intersects $ BC$ at $ Q.$ Prove that $ P$ lies on $ AB$ if and only if $ Q$ lies on segment $ EF$.