Found problems: 721
Let $ AD $ be the bisector of a triangle $ ABC $ $ (D \in BC) $ such that $ AB + AD = CD $ and $ AC + AD = BC $. Determine the measure of the angles of $ \vartriangle ABC $
On the sides $AB{}$ and $AC{}$ of the acute-angled triangle $ABC{}$ the points $M{}$ and $N{}$ are chosen such that $MN$ passes through the circumcenter of $ABC.$ Let $P{}$ and $Q{}$ be the midpoints of the segments $CM{}$ and $BN{}.$ Prove that $\angle POQ=\angle BAC.$
In the plane of a triangle $ABC,$ in its exterior$,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45^{\circ}$, $\angle BCP = \angle QCA = 30^{\circ}$, $\angle ABR = \angle RAB = 15^{\circ}$.
Prove that
[b]a.)[/b] $\angle QRP = 90\,^{\circ},$ and
[b]b.)[/b] $QR = RP.$
Consider a circle $C$ of center $O$ and its inner point $Q$, different from $O$. Where we must place a point $P$ on the circle $C$ so that the angle $\angle OPQ$ is the largest possible?
The electric train traveled from platform A to platform B in $X$ minutes ($0< X<60$). Find $X$ if it is known that as at the moment departure from A, and at the time of arrival at B, the angle between hourly and the minute hand was equal to $X$ degrees.
In the isosceles triangle $\vartriangle ABC$ is $AB = AC$. Let $D$ be the midpoint of the segment $BC$. The points $P$ and $Q$ are respectively on the lines $AD$ and $AB$ (with $Q \ne B$) so that $PQ = PC$. Show that $\angle PQC =\frac12 \angle A $
The square $ABCD$ and the equilateral triangle $MKL$ are located as shown in the figure. Find the angle $\angle PQD$.
[img]https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQKgjvzy1WhwkMJbcV_C0iveelYmm75FpaGlWgZ-Ap_uQUiegaKYafelo-J_3rMgKMgpMp5soYc1LVYLI8H4riC6R-f8eq2DiWTGGII08xQkwu7t2KVD4pKX4_IN-gC7DVRhdVZSjbaj2S/s1600/oral+moscow+geometry+2015+8.9+p2.png[/img]
The vertex $ A$ of the acute triangle $ ABC$ is equidistant from the circumcenter $ O$ and the orthocenter $ H.$ Determine all possible values for the measure of angle $ A.$
A polyhedron has $12$ faces and is such that:
[b][i](i)[/i][/b] all faces are isosceles triangles,
[b][i](ii)[/i][/b] all edges have length either $x$ or $y$,
[b][i](iii)[/i][/b] at each vertex either $3$ or $6$ edges meet, and
[b][i](iv)[/i][/b] all dihedral angles are equal.
Find the ratio $x/y.$
Let $O$ and $H$ be the circumcenter and the orthocenter of a triangle $ABC$. The line passing through the midpoint of $OH$ and parallel to $BC$ meets $AB$ and $AC$ at points $D$ and $E$. It is known that $O$ is the incenter of triangle $ADE$. Find the angles of $ABC$.
Consider an isosceles triangle $ABC$, where $AB = AC$. $D$ is a point on the $AC$ side and $P$ a point on the segment $BD$ so that the angle $\angle APC = 90^o$ and $ \angle ABP = \angle BCP $. Determine the ratio $AD: DC$.
What is the maximum number of acute interior angles a non-overlapping planar $2017$-gon can have?
Let $ABCD$ be a convex quadrilateral, and write $\alpha=\angle DAB$, $\beta=\angle ADB$, $\gamma=\angle ACB$, $\delta= \angle DBC$ and $\epsilon=\angle DBA$. Assuming that $\alpha<\pi/2$, $\beta+\gamma=\pi /2$, and $\delta+2\epsilon=\pi$, prove that $(DB+BC)^2=AD^2+AC^2$.
In triangle $ABC$, angle $A$ is equal to $a$ and angle $B$ is equal to $2a$. A circle with center at point $C$ of radius $CA$ intersects the line containing the bisector of the exterior angle at vertex $B$, at points $M$ and $N$. Find the angles of triangle $MAN$.
Let $ABC$ be an isosceles right-angled triangle, having the right angle at vertex $C$. Let us consider the line through $C$ which is parallel to $AB$ and let $D$ be a point on this line such that $AB = BD$ and $D$ is closer to $B$ than to $A$. Find the angle $\angle CBD$.
Points $M$ and $N$ are taken on the hypotenuse $AB$ of a right triangle $ABC$ so that $BC = BM$ and $AC = AN$. Prove that the angle $MCN$ is equal to $45$ degrees.
(Folklore)
Let $ABC$ be a triangle with $ \angle ACB = 90^o + \frac12 \angle ABC$ . The point $M$ is the midpoint of the side $BC$ . A circle with center at vertex $A$ intersects the line $BC$ at points $M$ and $D$. Prove that $MD = AB$.
Inside the parallelogram $ABCD$, point $M$ is chosen, and inside the triangle $AMD$, point $N$ is chosen in such a way that $$\angle MNA + \angle MCB =\angle MND + \angle MBC = 180^o.$$ Prove that lines $MN$ and $AB$ are parallel.
On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·
One square is inscribed in a circle, and another square is circumscribed around the same circle so that its vertices lie on the extensions of the sides of the first (see figure). Find the angle between the sides of these squares.
[img]https://3.bp.blogspot.com/-8eLBgJF9CoA/XTodHmW87BI/AAAAAAAAKY0/xsHTx71XneIZ8JTn0iDMHupCanx-7u4vgCK4BGAYYCw/s400/sharygin%2Boral%2B2017%2B10-11%2Bp1.png[/img]
A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.
In the plane, there is an acute angle $\angle AOB$ . Inside the angle points $C$ and $D$ are chosen so that $\angle AOC = \angle DOB$. From point $D$ the perpendicular on $OA$ intersects the ray $OC$ at point $G$ and from point C the perpendicular on $OB$ intersects the ray $OD$ at point $H$. Prove that the points $C, D, G$ and $H$ are conlyclic.
A car is driving along a straight highway at a speed of $60$ km per hour. Not far from the highway there is a parallel to him a $100$-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than $1100^o$
$ABC$ is a triangle. A circle through $A$ touches the side $BC$ at $D$ and intersects the sides $AB$ and $AC$ again at $E, F$ respectively. $EF$ bisects $\angle AFD$ and $\angle ADC = 80^o$. Find $\angle ABC$.
In the triangle $ABC$ on the sides $AB$ and $AC$ outward constructed equilateral triangles $ABD$ and $ACE$. The segments $CD$ and $BE$ intersect at point $F$. It turns out that point $A$ is the center of the circle inscribed in triangle $ DEF$. Find the angle $BAC$.
(Rozhkova Maria)