Found problems: 1065
Let $ ABC $ be an isosceles triangle with $ AC=BC $ . Take points $ D $ on side $AC$ and $E$ on side $BC$ and $ F $ the intersection of bisectors of angles $ DEB $ and $ADE$ such that $ F$ lies on side $AB$. Prove that $F$ is the midpoint of $AB$.
In triangle $ABC$ with $AB>BC$, $BM$ is a median and $BL$ is an angle bisector. The line through $M$ and parallel to $AB$ intersects $BL$ at point $D$, and the line through $L$ and parallel to $BC$ intersects $BM$ at point $E$. Prove that $ED$ is perpendicular to $BL$.
In convex quadrilateral $ ABCD$, the rays $ BA,CD$ meet at $ P$, and the rays $ BC,AD$ meet at $ Q$. $ H$ is the projection of $ D$ on $ PQ$. Prove that there is a circle inscribed in $ ABCD$ if and only if the incircles of triangles $ ADP,CDQ$ are visible from $ H$ under the same angle.
Given is an acute angled triangle $ \triangle ABC$ with side lengths $ a$, $ b$ and $ c$ (in an usual way) and circumcenter $ O$. Angle bisector of angle $ \angle BAC$ intersects circumcircle at points $ A$ and $ A_{1}$. Let $ D$ be projection of point $ A_{1}$ onto line $ AB$, $ L$ and $ M$ be midpoints of $ AC$ and $ AB$ , respectively.
(i) Prove that $ AD\equal{}\frac{1}{2}(b\plus{}c)$
(ii) If triangle $ \triangle ABC$ is an acute angled prove that $ A_{1}D\equal{}OM\plus{}OL$
Let $ABC$ be a triangle and let $O$ be its circumcentre. The internal and external bisectrices of the angle $BAC$ meet the line $BC$ at points $D$ and $E$, respectively. Let further $M$ and $L$ respectively denote the midpoints of the segments $BC$ and $DE$. The circles $ABC$ and $ALO$ meet again at point $N$. Show that the angles $BAN$ and $CAM$ are equal.
Let $\triangle ABC$ be the acute triangle such that $AB\ne AC.$ Let $V$ be the intersection of $BC$ and angle bisector of $\angle A.$ Let $D$ be the foot of altitude from $A$ to $BC.$ Let $E,F$ be the intersection of circumcircle of $\triangle AVD$ and $CA,AB$ respectively. Prove that the lines $AD, BE,CF$ is concurrent.
Let $ABC$ be an acute triangle with $AB = 3$ and $AC = 4$. Suppose that $AH,AO$ and $AM$ are the altitude, the bisector and the median derived from $A$, respectively. If $HO = 3 MO$, then the length of $BC$ is
[img]https://cdn.artofproblemsolving.com/attachments/e/c/26cc00629f4c0ab27096b8bdc562c56ff01ce5.png[/img]
A. $3$ B. $\frac72$ C. $4$ D. $\frac92$ E. $5$
Let $BF$ and $CN$ be the altitudes of the acute triangle $ABC$. Bisectors the angles $ACN$ and $ABF$ intersect at the point $T$. Find the radius of the circle circumscribed around the triangle $FTN$, if it is known that $BC = a$.
(Grigory Filippovsky)
Let $ABCD$ be a parallelogram such that $AB = 35$ and $BC = 28$. Suppose that $BD \perp BC$. Let $\ell_1$ be the reflection of $AC$ across the angle bisector of $\angle BAD$, and let $\ell_2$ be the line through $B$ perpendicular to $CD$. $\ell_1$ and $\ell_2$ intersect at a point $P$. If $PD$ can be expressed in simplest form as $\frac{m}{n}$, find $m + n$.
Given a convex pentagon $ABCDE$ such that $AE$ is parallel to $CD$ and $AB=BC$. Angle bisectors of angles $A$ and $C$ intersect at $K$. Prove that $BK$ and $AE$ are parallel.
Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$. Prove that $BR\parallel AC$.
(Here we always assume that an angle bisector is a ray.)
[i]Proposed by Sergey Berlov, Russia[/i]
In rectangle $ABCD$, $AB=20$ and $BC=10$. Let $E$ be a point on $\overline{CD}$ such that $\angle CBE=15^\circ$. What is $AE$?
$ \textbf{(A)}\ \dfrac{20\sqrt3}3\qquad\textbf{(B)}\ 10\sqrt3\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 11\sqrt3\qquad\textbf{(E)}\ 20 $
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 $ABC$.
Let $A_{1}$, $B_{1}$, $C_{1}$ be the feet of the altitudes of an acute-angled triangle $ABC$ issuing from the vertices $A$, $B$, $C$, respectively. Let $K$ and $M$ be points on the segments $A_{1}C_{1}$ and $B_{1}C_{1}$, respectively, such that $\measuredangle KAM = \measuredangle A_{1}AC$. Prove that the line $AK$ is the angle bisector of the angle $C_{1}KM$.
Let $ABCD$ be a parallelogram and point $M$ be the midpoint of $[AB]$ so that the quadrilateral $MBCD$ is cyclic. If $N$ is the point of intersection of the lines $DM$ and $BC$, and $P \in BC$, then prove that the ray $(DP$ is the angle bisector of $\angle ADM$ if and only if $PC = 4BC$.
In triangle $ABC$, the angle bisectors are $BE$ and $CF$ (where $E, F$ are on the sides of the triangle), and their intersection point is $I$. Point $N$ lies on the circumcircle of $AEF$, and the angle $\angle IAN$ is right. The circumcircle of $AEF$ meets the line $NI$ a second time at the point $L$. Show that the circumcenter of $AIL$ lies on line $BC$.
Let $ABCD$ to be a quadrilateral inscribed in a circle $\Gamma$. Let $r$ and $s$ to be the tangents to $\Gamma$ through $B$ and $C$, respectively, $M$ the intersection between the lines $r$ and $AD$ and $N$ the intersection between the lines $s$ and $AD$. After all, let $E$ to be the intersection between the lines $BN$ and $CM$, $F$ the intersection between the lines $AE$ and $BC$ and $L$ the midpoint of $BC$. Prove that the circuncircle of the triangle $DLF$ is tangent to $\Gamma$.
In $\vartriangle ABC$, the bisector of $\angle B$ meets $AC$ at $D$ and the bisector of $\angle C$ meets $AB$ at $E$. These bisectors intersect at $O$ and $OD = OE$. If $AD \ne AE$, prove that $\angle A = 60^o$.
Let $ABC$ be a triangle, $I$ its incenter, and $D$ a point on the arc $BC$ of the circumcircle of $ABC$ not containing $A$. The bisector of the angle $\angle ADB$ intesects the segment $AB$ at $E$. The bisector of the angle $\angle CDA$ intesects the segment $AC$ at $F$. Prove that the points $E, F,I$ are collinear.
(Malik Talbi)
In triangle $ABC$, point$ I$ is incenter , $I_a$ is the $A$-excenter. Let $K$ be the intersection point of the $BC$ with the external bisector of the angle $BAC$, and $E$ be the midpoint of the arc $BAC$ of the circumcircle of triangle $ABC$. Prove that $K$ is the orthocenter of triangle $II_aE$.
Let angle bisectors of angles $\angle BAC$ and $\angle ABC$ of triangle $ABC$ intersect sides $BC$ and $AC$ in points $D$ and $E$, respectively. Let points $F$ and $G$ be foots of perpendiculars from point $C$ on lines $AD$ and $BE$, respectively. Prove that $FG \mid \mid AB$
Let $C_1$ and $C_2$ be two concentric circles with $C_1$ inside $C_2$. Let $P_1$ and $P_2$ be two points on $C_1$ that are not diametrically opposite. Extend the segment $P_1P_2$ past $P_2$ until it meets the circle $C_2$ in $Q_2$. The tangent to $C_2$ at $Q_2$ and the tangent to $C_1$ at $P_1$ meet in a point $X$. Draw from X the second tangent to $C_2$ which meets $C_2$ at the point $Q_1$. Show that $P_1X$ bisects angle $Q_1P_1Q_2$.
Show that in a right-angled triangle the bisector of the right angle divides into equal parts the angle between the altitude and the median, drawn from the same vertex.
The feet of the perpendiculars from the intersection point of the diagonals of a convex cyclic quadrilateral to the sides form a quadrilateral $q$. Show that the sum of the lengths of each pair of opposite sides of $q$ is equal.
In the convex quadrilateral $ABCD$ we have that $\angle BCD = \angle ADC \ge 90 ^o$. The bisectors of $\angle BAD$ and $\angle ABC$ intersect in $M$. Prove that if $M \in CD$, then $M$ is the middle of $CD$.