Found problems: 1065
Let $D$ be a point inside $\triangle ABC$ such that $m(\widehat{BAD})=20^{\circ}$, $m(\widehat{DAC})=80^{\circ}$, $m(\widehat{ACD})=20^{\circ}$, and $m(\widehat{DCB})=20^{\circ}$.
$m(\widehat{ABD})= ?$
$ \textbf{(A)}\ 5^{\circ} \qquad \textbf{(B)}\ 10^{\circ} \qquad \textbf{(C)}\ 15^{\circ} \qquad \textbf{(D)}\ 20^{\circ} \qquad \textbf{(E)}\ 25^{\circ}$
A quadrilateral $ABCD$ is such that the sides $AB$ and $DC$ are parallel, and $|BC| =|AB| + |CD|$. Prove that the angle bisectors of the angles $\angle ABC$ and $\angle BCD$ intersect at right angles on the side $AD$.
Let $M$ be an arbitrary point on side $BC$ of triangle $ABC$. $W$ is a circle which is tangent to $AB$ and $BM$ at $T$ and $K$ and is tangent to circumcircle of $AMC$ at $P$. Prove that if $TK||AM$, circumcircles of $APT$ and $KPC$ are tangent together.
For a nonisosceles triangle $ABC$, consider the altitude from vertex $A$ and two bisectrices from remaining vertices. Prove that the circumcircle of the triangle formed by these three lines touches the bisectrix from vertex $A$.
On the sides $BC$ and $AC$ of the isosceles triangle $ABC$ ($AB = BC$), points $E$ and $D$ are marked, respectively, so that $DE \parallel AB$. On the extendsion of side $CB$ beyond the point $B$, point $K$ was arbitrarily marked. Let $P$ be the intersection point of the lines $AB$ and $KD$. Let $Q$ be the intersection point of the lines $AK$ and $DE$. Prove that $CA$ is the bisector of angle $\angle PCQ$.
Let $\vartriangle ABC$ be a triangle with $\angle ABC > \angle BCA \ge 30^o$ . The angle bisectors of $\angle ABC$ and $\angle BCA$ intersect $CA$ and $AB$ at $D$ and $E$ respectively, and $BD$ and $CE$ intersect at $P$. Suppose that $P D = P E$ and the incircle of $\vartriangle ABC$ has unit radius. What is the maximum possible length of $BC$?
Let $\Gamma$ be the circumcircle of the acute triangle $ABC$. The angle bisector of angle $ABC$ intersects $AC$ in the point $B_1$ and the short arc $AC$ of $\Gamma$ in the point $P$. The line through $B_1$ perpendicular to $BC$ intersects the short arc $BC$ of $\Gamma$ in $K$. The line through $B$ perpendicular to $AK$ intersects $AC$ in $L$. Prove that $K, L$ and $P$ lie on a line.
Consider the triangle $ABC$, where $\angle B= 30^o, \angle C = 15^o$, and $M$ is the midpoint of the side $[BC]$. Let point $N \in (BC)$ be such that $[NC] = [AB]$. Show that $[AN$ is the angle bisector of $MAC$
Let $ABC$ be an acute-angled triangle with $AB \ne BC$ and radius $k$. Let $P$ and $Q$ be the points of intersection of $k$ with the internal bisector and the external bisector of $\angle CBA$ respectively. Let $D$ be the intersection of $AC$ and $PQ$. Find the ratio $AD: DC$.
Let $ABC$ be an acute-angled triangle in which $|AB| < |AC|$ and let circle $k$ with center $O$ be its circumscribed circle. Let $P$ and $Q$ be points on sides $\overline{BC}$ and $\overline{AB}$ respectively such that $AQPO$ is a parallelogram. Let $K$ and $L$ be the intersections of the perpendicular bisector of $\overline{OP}$ with circle $k$, where $K$ is on the shorter arc $AB$. Let $M$ be the second intersection of the line $KQ$ with the circle $k$. Prove that the point $A$ belongs to the bisector of the angle $\angle QLM$.
Given triangle $ ABC$ of area 1. Let $ BM$ be the perpendicular from $ B$ to the bisector of angle $ C$. Determine the area of triangle $ AMC$.
Let $ABC$ be an acute triangle such that $AB<AC$. Let $\Omega$ be its circumcircle, $O$ be its circumcenter, and $l$ be the internal angle bisector of $\angle BAC$. Suppose that the tangents to $\Omega$ at $B$ and $C$ intersect at $X$. Let $\omega$ be a circle whose center is $X$ and passes $B$, and $Y$ be the intersection of $l$ and $\omega$ which is chosen inside $\triangle ABC$. Let $D,E$ be the projections of $Y$ onto $AB,AC$, respectively. $OY$ meets $BC$ at $Z$. $ZD,ZE$ meet $l$ at $P,Q$, respectively. Prove that $BQ$ and $CP$ are parallel.
An equilateral triangle with side length $6$ has a square of side length $6$ attached to each of its edges as shown. The distance between the two farthest vertices of this figure (marked $A$ and $B$ in the figure) can be written as $m + \sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$.
[asy]
draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle);
draw((1,0)--(1+sqrt(3)/2,1/2)--(1/2+sqrt(3)/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2));
draw((0,0)--(-sqrt(3)/2,1/2)--(-sqrt(3)/2+1/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2));
dot((-sqrt(3)/2+1/2,1/2+sqrt(3)/2));
label("A", (-sqrt(3)/2+1/2,1/2+sqrt(3)/2), N);
draw((1,0)--(1,-1)--(0,-1)--(0,0));
dot((1,-1));
label("B", (1,-1), SE);
[/asy]
Let $[ABC]$ be a triangle, $D$ be the orthogonal projection of $B$ on the bisector of $\angle ACB$ and $E$ the orthogonal projection of $C$ on the bisector of $\angle ABC$ . Prove that $DE$ intersects the sides $[AB]$ and $[AC]$ at the touchpoints of the circle inscribed in the triangle $[ABC]$.
The extension of the bisector of angle $A$ of triangle $ABC$ intersects with the circumscribed circle of this triangle at point $W$. A straight line is drawn through $W$, which is parallel to side $AB$ and intersects sides $BC$ and $AC$ , at points $N$ and $K$, respectively. Prove that the line $AW$ is tangent to the circumscribed circle of $\vartriangle CNW$.
(Sergey Yakovlev)
We consider a square $ABCD$ and a point $E$ on the segment $CD$. The bisector of $\angle EAB$ cuts the segment $BC$ in $F$. Prove that $BF + DE = AE$.
Let $ABC$ be a triangle and $D$ be a point on $BC$ such that $AB+BD=AC+CD$. The line $AD$ intersects the incircle of triangle $ABC$ at $X$ and $Y$ where $X$ is closer to $A$ than $Y$ i. Suppose $BC$ is tangent to the incircle at $E$, prove that:
1) $EY$ is perpendicular to $AD$;
2) $XD=2IM$ where $I$ is the incentre and $M$ is the midpoint of $BC$.
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 $CD=BE$.
Let $AK$ and $AT$ be the bisector and the median of an acute-angled triangle $ABC$ with $AC > AB$. The line $AT$ meets the circumcircle of $ABC$ at point $D$. Point $F$ is the reflection of $K$ about $T$. If the angles of $ABC$ are known, find the value of angle $FDA$.
On the side $BC$ of the triangle $ABC$ a point $D$ different from $B$ and $C$ is chosen so that the bisectors of the angles $ACB$ and $ADB$ intersect on the side $AB$. Let $D'$ be the symmetrical point to $D$ with respect to the line $AB$. Prove that the points $C, A$ and $D'$ are on the same line.
In triangle $ABC, AH$ is an altitude ($H$ is on $BC$) and $BE$ is a bisector ($E$ is on $AC$) . We are given that angle $BEA$ equals $45^o$ .Prove that angle $EHC$ equals $45^o$ .
(I. Sharygin , Moscow)
In $\triangle ABC$ the median $AM$ is drawn. The foot of perpendicular from $B$ to the angle bisector of $\angle BMA$ is $B_1$ and the foot of perpendicular from $C$ to the angle bisector of $\angle AMC$ is $C_1.$ Let $MA$ and $B_1C_1$ intersect at $A_1.$ Find $\frac{B_1A_1}{A_1C_1}.$
In triangle $ABC$ the angle bisector $AK$ is perpendicular on the median is $CL$. Prove that in the triangle $BKL$ also one of angle bisectors are perpendicular to one of the medians.
Cyclic quadrilateral $ABCD$ has circumcircle $(O)$. Points $M$ and $N$ are the midpoints of $BC$ and $CD$, and $E$ and $F$ lie on $AB$ and $AD$ respectively such that $EF$ passes through $O$ and $EO=OF$. Let $EN$ meet $FM$ at $P$. Denote $S$ as the circumcenter of $\triangle PEF$. Line $PO$ intersects $AD$ and $BA$ at $Q$ and $R$ respectively. Suppose $OSPC$ is a parallelogram. Prove that $AQ=AR$.
The point $M$ is the center of a regular pentagon $ABCDE$. The point $P$ is an inner point of the line segment $[DM]$. The circumscribed circle of triangle $\vartriangle ABP$ intersects the side $[AE]$ at point $Q$ (different from $A$). The perpendicular from $P$ on $CD$ intersects the side $[AE] $ at point $S$. Prove that $PS$ is the bisector of $\angle APQ$.