Found problems: 1065
Consider an isosceles triangle $KL_1L_2$ with $|KL_1|=|KL_2|$ and let $KA, L_1B_1,L_2B_2$ be its angle bisectors. Prove that $\cos \angle B_1AB_2 < \frac35$
A trapezoid $ABCD$ is given with $BC // AD$. Assume that the bisectors of the angles $BAD$ and $CDA$ intersect on the perpendicular bisector of the line segment $BC$. Prove that $|AB|= |CD|$ or $|AB| +|CD| =|AD|$.
Let $\Gamma$ be a circle with radius $r$. Let $A$ and $B$ be distinct points on $\Gamma$ such that $AB < \sqrt{3}r$. Let the circle with centre $B$ and radius $AB$ meet $\Gamma$ again at $C$. Let $P$ be the point inside $\Gamma$ such that triangle $ABP$ is equilateral. Finally, let the line $CP$ meet $\Gamma$ again at $Q$.
Prove that $PQ = r$.
Let $ABC$ be a triangle with incenter $I$. Suppose the reflection of $AB$ across $CI$ and the reflection of $AC$ across $BI$ intersect at a point $X$. Prove that $XI$ is perpendicular to $BC$.
It is given isosceles triangle $ABC$ ($AB=AC$) such that $\angle BAC=108^{\circ}$. Angle bisector of angle $\angle ABC$ intersects side $AC$ in point $D$, and point $E$ is on side $BC$ such that $BE=AE$. If $AE=m$, find $ED$
In triangle $ABC$, let $P$ and $R$ be the feet of the perpendiculars from $A$ onto the external and internal bisectors of $\angle ABC$, respectively; and let $Q$ and $S$ be the feet of the perpendiculars from $A$ onto the internal and external bisectors of $\angle ACB$, respectively. If $PQ = 7, QR = 6$ and $RS = 8$, what is the area of triangle $ABC$?
In parallelogram $ABCD$, points $M$ and $N$ are selected on sides $AB$ and $BC$ respectively so that $AM = NC$, $Q$ is the intersection point of segments $AN$ and $CM$. Prove that $DQ$ is the bisector of angle $D$.
In a non equilateral triangle $ABC$ the sides $a,b,c$ form an arithmetic progression. Let $I$ be the incentre and $O$ the circumcentre of the triangle $ABC$. Prove that
(1) $IO$ is perpendicular to $BI$;
(2) If $BI$ meets $AC$ in $K$, and $D$, $E$ are the midpoints of $BC$, $BA$ respectively then $I$ is the circumcentre of triangle $DKE$.
Triangle $ABC$ has points $A$ at $\left(0,0\right)$, $B$ at $\left(9,12\right)$, and $C$ at $\left(-6,8\right)$ in the coordinate plane. Find the length of the angle bisector of $\angle{BAC}$ from $A$ to where it intersects $BC$.
[i]2017 CCA Math Bonanza Lightning Round #1.3[/i]
Let an acute-angled triangle $ABC$ with $AB<AC<BC$, inscribed in circle $c(O,R)$. The angle bisector $AD$ meets $c(O,R)$ at $K$. The circle $c_1(O_1,R_1)$(which passes from $A,D$ and has its center $O_1$ on $OA$) meets $AB$ at $E$ and $AC$ at $Z$. If $M,N$ are the midpoints of $ZC$ and $BE$ respectively, prove that:
[b]a)[/b]the lines $ZE,DM,KC$ are concurrent at one point $T$.
[b]b)[/b]the lines $ZE,DN,KB$ are concurrent at one point $X$.
[b]c)[/b]$OK$ is the perpendicular bisector of $TX$.
In a triangle $ABC$ the point $D$ is the intersection of the interior angle bisector of $\angle BAC$ and side $BC$. Let $P$ be the second intersection point of the exterior angle bisector of $\angle BAC$ with the circumcircle of $\angle ABC$. A circle through $A$ and $P$ intersects line segment $BP$ internally in $E$ and line segment $CP$ internally in $F$. Prove that $\angle DEP = \angle DFP$.
Given is $\triangle ABC$ such that $\angle A = 57^o, \angle B = 61^o$ and $\angle C = 62^o$. Which segment is longer: the angle bisector through $A$ or the median through $B$?
(N.Beluhov)
A convex quadrangle $ABCD$ is given. Let $I$ and $J$ be the circles of circles inscribed in the triangles $ABC$ and $ADC$, respectively, and $I_a$ and $J_a$ are the centers of the excircles circles of triangles $ABC$ and $ADC$, respectively (inscribed in the angles $BAC$ and $DAC$, respectively). Prove that the intersection point $K$ of the lines $IJ_a$ and $JI_a$ lies on the bisector of the angle $BCD$.
Points $C_0$ and $B_0$ are the respective midpoints of sides $AB$ and $AC$ of a non-isosceles acute triangle $ABC$, $O$ is its circumscenter and $H$ is the orthocenter. Lines $BH$ and $OC_0$ intersect at $P$, while lines $CH$ and $OB_0$ intersect at $Q$. $OPHQ$ is rhombus. Prove that points $A$, $P$ and $Q$ are collinear. (Author: L. Emelyanov)
Let $ABC$ be a triangle, and $K$ and $L$ be points on $AB$ such that $\angle ACK = \angle KCL = \angle LCB$. Let $M$ be a point in $BC$ such that $\angle MKC = \angle BKM$. If $ML$ is the angle bisector of $\angle KMB$, find $\angle MLC$.
Let $ABC$ be a triangle with $AB < AC$ and incentre $I$. Let $E$ be the point on the side $AC$ such that $AE = AB$. Let $G$ be the point on the line $EI$ such that $\angle IBG = \angle CBA$ and such that $E$ and $G$ lie on opposite sides of $I$.
Prove that the line $AI$, the line perpendicular to $AE$ at $E$, and the bisector of the angle $\angle BGI$ are concurrent.
Let $ABC$ be triangle such that $|AB| = 5$, $|BC| = 9$ and $|AC| = 8$. The angle bisector of $\widehat{BCA}$ meets $BA$ at $X$ and the angle bisector of $\widehat{CAB}$ meets $BC$ at $Y$. Let $Z$ be the intersection of lines $XY$ and $AC$. What is $|AZ|$?
$
\textbf{a)}\ \sqrt{104}
\qquad\textbf{b)}\ \sqrt{145}
\qquad\textbf{c)}\ \sqrt{89}
\qquad\textbf{d)}\ 9
\qquad\textbf{e)}\ 10
$
In the triangle $ABC$, we have that $\angle BAC$ is acute. Let $\Gamma$ be the circle that passes through $A$ and is tangent to the side $BC$ at $C$. Let $M$ be the midpoint of $BC$ and let $D$ be the other point of intersection of $\Gamma$ with $AM$. If $BD$ cuts back to$ \Gamma$ at $E$, show that $AC$ is the bisector of $\angle BAE$.
Let $ABC$ be a triangle and let $N$ and $M$ be the midpoints of $AB$ and $CA$, respectively. Let $H$ be the foot of altitude from $A$. The circumcircle of $ABH$ intersects $MN$ at $P$, with $P$ and $M$ on the same side relative to $N$, and the circumcircle of $ACH$ intersects $MN$ at $Q$, with $Q$ and $N$ on the same side relative to $M$. $BP$ and $CQ$ intersect at $X$. Prove that $AX$ is the angle bisector of $\angle CAB$.
Let $ C$ be a circle in plane. Let $ C_1$ and $ C_2$ be nonintersecting circles touching $ C$ internally at points $ A$ and $ B$ respectively. Let $ t$ be a common tangent of $ C_1$ and $ C_2$ touching them at points $ D$ and $ E$ respectively, such that both $ C_1$ and $ C_2$ are on the same side of $ t$. Let $ F$ be the point of intersection of $ AD$ and $ BE$. Show that $ F$ lies on $ C$.
The points $M$ and $N$ are chosen on the angle bisector $AL$ of a triangle $ABC$ such that $\angle ABM=\angle ACN=23^{\circ}$. $X$ is a point inside the triangle such that $BX=CX$ and $\angle BXC=2\angle BML$. Find $\angle MXN$.
A line $r$ contains the points $A$, $B$, $C$, $D$ in that order. Let $P$ be a point not in $r$ such that $\angle{APB} = \angle{CPD}$. Prove that the angle bisector of $\angle{APD}$ intersects the line $r$ at a point $G$ such that:
$\frac{1}{GA} + \frac{1}{GC} = \frac{1}{GB} + \frac{1}{GD}$
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 $AZ \perp BC$.
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?
In triangle $ABC$, $AB=125,AC=117$, and $BC=120$. The angle bisector of angle $A$ intersects $\overline{BC}$ at point $L$, and the angle bisector of angle $B$ intersects $\overline{AC}$ at point $K$. Let $M$ and $N$ be the feet of the perpendiculars from $C$ to $\overline{BK}$ and $\overline{AL}$, respectively. Find $MN$.