Found problems: 721
Show that any two points lying inside a regular $ n\minus{}$gon $ E$ can be joined by two circular arcs lying inside $ E$ and meeting at an angle of at least $ \left(1 \minus{} \frac{2}{n} \right) \cdot \pi.$
Point $O$ is the intersection of the perpendicular bisectors of the sides of the triangle $\vartriangle ABC$ . Let $D$ be the intersection of the line $AO$ with the segment $[BC]$. Knowing that $OD = BD = \frac 13 BC$, find the measures of the angles of the triangle $\vartriangle ABC$.
Inside the quadrilateral $ABCD$ marked a point $O$ such that $\angle OAD+ \angle OBC = \angle ODA + \angle OCB = 90^o$. Prove that the centers of the circumscribed circles around triangles $OAD$ and $OBC$ as well as the midpoints of the sides $AB$ and $CD$ lie on one circle.
(Anton Trygub)
A circle is inscribed in an angle with vertex $O$, touching its sides at points $M$ and $N$. On an arc $MN$ nearest to point $O$, an arbitrary point $P$ is selected. At point $P$, a tangent is drawn to the circle $P$, intersecting the sides of the angle at points $A$ and $B$. Prove that that the length of the segment $AB$ is the smallest when $P$ is its midpoint.
In an isosceles right-angled triangle $ABC$, the right angle is at $A$. $D$ lies so on the side $BC$ that $2CD = DB$. Let $E$ be the projection of $B$ onto $AD$. What is the measure fof angle $\angle CED $?
In an acute-angled triangle $ABC$ on the side $AC$, point $P$ is chosen in such a way that $2AP = BC$. Points $X$ and $Y$ are symmetric to $P$ with respect to vertices $A$ and $C$, respectively. It turned out that $BX = BY$. Find $\angle BCA$.
Segment equal to median $AA_0$ of triangle $ABC$ is drawn from point $A_0$ perpendicular to side $BC$ to the outer side of the triangle. Let's denote the second end of the constructed segment as $A_1$. Points $B_1$ and $C_1$ are constructed similarly. Find the angles of triangle $A_1B_1C_1$ if the angles of the triangle $ABC$ are $30^o$, $30^o$ and $120^o$.
[hide=original wording]Медиану AA0 треугольника ABC отложили от точки A0 перпендикулярно стороне BC во внешнюю сторону треугольника. Обозначим второй конец построенного отрезка через A1. Аналогично строятся точки B1 и C1. Найдите углы треугольника A1B1C1, если углы треугольника ABC равны 30^o, 30^o и 120^o.[/hide]
Let $ABCD$ be a trapezoid of parallel bases $ BC$ and $AD$. If $\angle CAD = 2\angle CAB, BC = CD$ and $AC = AD$, determine all the possible values of the measure of the angle $\angle CAB$.
In an acute-angled triangle $ABC$ on the side $AC$, point $P$ is chosen in such a way that $2AP = BC$. Points $X$ and $Y$ are symmetric to $P$ with respect to vertices $A$ and $C$, respectively. It turned out that $BX = BY$. Find $\angle BCA$.
The triangle $ABC$ is isosceles with $AB=BC$. The point F on the side $[BC]$ and the point $D$ on the side $AC$ are the feets of the the internals bisectors drawn from $A$ and altitude drawn from $B$ respectively so that $AF=2BD$. Fine the measure of the angle $ABC$.
A non-right triangle $A_1A_2A_3$ is given. Circles $C_1$ and $C_2$ are tangent at $A_3, C_2$ and $C_3$ are tangent at $A_1$, and $C_3$ and $C_1$ are tangent at $A_2$. Points $O_1,O_2,O_3$ are the centers of $C_1, C_2, C_3$, respectively. Supposing that the triangles $A_1A_2A_3$ and $O_1O_2O_3$ are similar, determine their angles.
Let $ABC$ be a non-isosceles triangle with circumcenter $O$, incenter $I$, and orthocenter $H$. Prove that angle $\angle OIH$ is obtuse.
Let $ O $ be the center of the inscribed circle of the triangle $ ABC $, tangent to the side of $ BC $. Let $ M $ be the midpoint of $ AC $, and $ P $ be the intersection point of $ MO $ and $ BC $. Prove that $ AB = BP $ if $ \angle BAC = 2 \angle ACB $.
Segment $AB$ on the surface of the cube is the shortest polyline on the surface that connects $A$ and $B$. Triangle $ABC$ consisted of such segments $AB, BC,CA$. What may be the sum of angles of such triangle if none of the vertex is on the edge of the cube ?
The triangle $ABC$ is given. Let $P_1$ and $P_2$ be points on the side $AB$ such that $P_2$ lies on the segment $BP_1$ and $AP_1 = BP_2$. Similarly, $Q_1$ and $Q_2$ are points on the side $BC$ such that $Q_2$ lies on the segment $BQ_1$ and $BQ_1 = CQ_2$. The segments $P_1Q_2$ and $P_2Q_1$ intersect at the point $R{}$, and the circles $P_1P_2R$ and $Q_1Q_2R$ intersect a second time at the point $S{}$ lying inside the triangle $P_1Q_1R$. Let $M{}$ be the midpoint of the segment $AC$. Prove that the angles $P_1RS$ and $Q_1RM$ are equal.
Find the angle $BCA$ in the quadrilateral of the figure.
[img]https://cdn.artofproblemsolving.com/attachments/0/2/974e23be54125cde8610a78254b59685833b5b.png[/img]
Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.
In triangle $ABC$, angle $A$ is twice as large as angle $B$. $AB = 3$ and $AC = 2$. Calculate $BC$.
The point $M$ , inside $\vartriangle ABC$, satisfies the conditions that $\angle BMC = 90^o +\frac12 \angle BAC$ and that the line $AM$ contains the centre of the circumscribed circle of $\vartriangle BMC$. Prove that $M$ is the centre of the inscribed circle of $\vartriangle ABC$.
Given triangle $ABC$ such that $AB- BC = \frac{AC}{\sqrt2}$ . Let $M$ be the midpoint of $AC$, and $N$ be the foot of the angle bisector from $B$. Prove that $\angle BMC + \angle BNC = 90^o$.
(A.Akopjan)
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]
In the triangle $ABC$, angle $C$ is four times smaller than each of the other two angle The altitude $AK$ and the angle bisector $AL$ are drawn from the vertex of the angle $A$. It is known that the length of $AL$ is equal to $\ell$. Find the length of the segment $LK$.
(Gryhoriy Filippovskyi)
The quadrilateral $ABCD$ is known to have $BC = CD = AC$, and the angle $\angle ABC= 70^o$. Calculate the degree measure of the angle $\angle ADB$.
(Alexey Panasenko)
Let $\Gamma $ be the circumcircle of triangle $ABC$. A circle $\Omega$ is tangent to the line segment $AB$ and is tangent to $\Gamma$ at a point lying on the same side of the line $AB$ as $C$. The angle bisector of $\angle BCA$ intersects $\Omega$ at two different points $P$ and $Q$.
Prove that $\angle ABP = \angle QBC$.
In an acute triangle $ABC$ with an angle $\angle ACB =75^o$, altitudes $AA_3,BB_3$ intersect the circumscribed circle at points $A_1,B_1$ respectively. On the lines $BC$ and $CA$ select points $A_2$ and $B_2$ respectively suchthat the line $B_1B_2$ is parallel to the line $BC$ and the line $A_1A_2$ is parallel to the line $AC$ . Let $M$ be the midpoint of the segment $A_2B_2$. Find in degrees the measure of the angle $\angle B_3MA_3$.