Found problems: 721
Let $A_1, A_2, . . . , A_{2021}$ be $2021$ points on the plane, no three collinear and $$\angle A_1A_2A_3 + \angle A_2A_3A_4 +... + \angle A_{2021}A_1A_2 = 360^o,$$ in which by the angle $\angle A_{i-1}A_iA_{i+1}$ we mean the one which is less than $180^o$ (assume that $A_{2022} =A_1$ and $A_0 = A_{2021}$). Prove that some of these angles will add up to $90^o$.
[i]Proposed by Morteza Saghafian - Iran[/i]
Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.
In the right-angled triangle $ABC$ with a right angle at $C$, the side $BC$ is longer than the side $AC$. The perpendicular bisector of $AB$ intersects the line $BC$ at point $D$ and the line $AC$ at point $E$. The segments $DE$ has the same length as the side $AB$. Find the measures of the angles of the triangle $ABC$.
(R. Henner, Vienna)
An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.
Equal line segments are marked in triangle $ABC$. Find its angles.
[img]https://cdn.artofproblemsolving.com/attachments/0/2/bcb756bba15ba57013f1b6c4cbe9cc74171543.png[/img]
Let $ABCD$ be a square, $E$ a point on the side $CD$, and $F$ a point inside the square such that that triangle $BFE$ is isosceles and $\angle BFE = 90^o$ . If $DF=DE$, find the measure of angle $\angle FDE$.
Let $M$ be the midpoint of segment $[AB]$ in triangle $\triangle ABC$. Let $X$ and $Y$ be points such that $\angle{BAX}=\angle{ACM}$ and $\angle{BYA}=\angle{MCB}$. Both points, $X$ and $Y$, are on the same side as $C$ with respect to line $AB$.
Show that the rays $[AX$ and $[BY$ intersect on line $CM$.
Inside the acute-angled triangle $ABC$, the points $P{}$ and $Q{}$ are chosen so that $\angle ACP = \angle BCQ$ and $\angle CBP =\angle ABQ$. The point $Z{}$ is the projection of $P{}$ onto the line $BC$. The point $Q'$ is symmetric to $Q{}$ with respect to $Z{}$. The points $K{}$ and $L{}$ are chosen on the rays $AB$ and $AC$ respectively, so that $Q'K \parallel QC$ and $Q'L \parallel QB$. Prove that $\angle KPL=\angle BPC$.
Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.
Two circles with radii $1$ and $2$ have a common center at the point $O$. The vertex $A$ of the regular triangle $ABC$ lies on the larger circle, and the middpoint of the base $CD$ lies on the smaller one. What can the angle $BOC$ be equal to?
In the quadrilateral $ABCD$, the length of the sides $AB$ and $BC$ is equal to $1, \angle B= 100^o , \angle D= 130^o$ . Find the length of $BD$.
The pyramid $VABCD$ has base the rectangle ABCD, and the side edges are congruent. Prove that the plane $(VCD)$ forms congruent angles with the planes $(VAC)$ and $(BAC)$ if and only if $\angle VAC = \angle BAC $.
The convex quadrilateral $ABCD$ is has angle $A$ equal to $60^o$ , angle bisector of $A$ the diagonal $AC$ and $\angle ACD= 40^o$ and $\angle ACB = 120^o$. Inside the quadrilateral the point $P$ lies such that $\angle PDA = 40^o$ and $\angle PBA = 10^o$;
a) Find the angle $\angle DPB$?
b) Prove that $P$ lies on the diagonal $AC$.
The lengths, $a$ and $b$, of two sides of a triangle are known.
(a) What length should the third side be, in order for the largest angle of the triangle to be of the least possible value?
(b) What length should the third side be in order for the smallest angle of the triangle to be of the greatest possible value?
Let $ABC$ be a triangle and $D$ a point inside the triangle, located on the median of $A$. Prove that if $\angle BDC = 180^o - \angle BAC$, then $AB \cdot CD = AC \cdot BD$.
In an acute-angled triangle, each angle is an integral number of degrees, and the smallest angle is one-fifth of the largest one. Find these angles.
(G Galperin)
Let $ABCD$ be a rectangle with $AB\ge BC$ Point $M$ is located on the side $(AD)$, and the perpendicular bisector of $[MC]$ intersects the line $BC$ at the point $N$. Let ${Q} =MN\cup AB$ . Knowing that $\angle MQA= 2\cdot \angle BCQ $, show that the quadrilateral $ABCD$ is a square.
Let a triangle $ABC$ be given. Two circles passing through $A$ touch $BC$ at points $B$ and $C$ respectively. Let $D$ be the second common point of these circles ($A$ is closer to $BC$ than $D$). It is known that $BC = 2BD$. Prove that $\angle DAB = 2\angle ADB.$
For any given triangle $A_0B_0C_0$ consider a sequence of triangles constructed as follows:
a new triangle $A_1B_1C_1$ (if any) has its sides (in cm) that equal to the angles of $A_0B_0C_0$ (in radians). Then for $\vartriangle A_1B_1C_1$ consider a new triangle $A_2B_2C_2$ (if any) constructed in the similar พay, i.e., $\vartriangle A_2B_2C_2$ has its sides (in cm) that equal to the angles of $A_1B_1C_1$ (in radians), and so on.
Determine for which initial triangles $A_0B_0C_0$ the sequence never terminates.
The heights of a triangular pyramid intersect at one point. Prove that all flat angles at any vertex of the surface are either acute, or right, or obtuse.
Let $\alpha,\beta,\gamma$ be angles of a triangle. Two of them can be expressed using an auxiliary angle $\varphi$ in a way that $$\alpha=\varphi+\frac\pi4,\quad\beta=\pi-3\varphi.$$ Show that $\alpha>\gamma.$
What is the greatest number of rays in space beginning at one point and forming pairwise obtuse angles?
Inside an equilateral triangle $ABC$ one constructs points $P, Q$ and $R$ such that
\[\angle QAB = \angle PBA = 15^\circ,\\ \angle RBC = \angle QCB = 20^\circ,\\ \angle PCA = \angle RAC = 25^\circ.\]
Determine the angles of triangle $PQR.$
Let $ABCD$ be a square. The points $N$ and $P$ are chosen on the sides $AB$ and $AD$ respectively, such that $NP=NC$. The point $Q$ on the segment $AN$ is such that that $\angle QPN=\angle NCB$. Prove that $\angle BCQ=\frac{1}{2}\angle AQP$.
The sides $BC$ and $AD$ of a quadrilateral $ABCD$ are parallel and the diagonals intersect in $O$. For this quadrilateral $|CD| =|AO|$ and $|BC| = |OD|$ hold. Furthermore $CA$ is the angular bisector of angle $BCD$. Determine the size of angle $ABC$.
[asy]
unitsize(1 cm);
pair A, B, C, D, O;
D = (0,0);
B = 3*dir(180 + 72);
C = 3*dir(180 + 72 + 36);
A = extension(D, D + (1,0), C, C + dir(180 - 36));
O = extension(A, C, B, D);
draw(A--B--C--D--cycle);
draw(B--D);
draw(A--C);
dot("$A$", A, N);
dot("$B$", B, SW);
dot("$C$", C, SE);
dot("$D$", D, N);
dot("$O$", O, E);
[/asy]
Attention: the figure is not drawn to scale.