Found problems: 721
a) Let $UVW$ , $U'V'W'$ be two triangles such that $ VW = V'W' , UV = U'V' , \angle WUV = \angle W'U'V'.$
Prove that the angles $\angle VWU , \angle V'W'U'$ are equal or supplementary.
b) $ABC$ is a triangle where $\angle A$ is [b]obtuse[/b]. take a point $P$ inside the triangle , and extend $AP,BP,CP$ to meet the sides $BC,CA,AB$ in $K,L,M$ respectively. Suppose that $PL = PM .$
1) If $AP$ bisects $\angle A$ , then prove that $AB = AC$ .
2) Find the angles of the triangle $ABC$ if you know that $AK,BL,CM$ are angle bisectors of the triangle $ABC$ and that $2AK = BL$.
The median $AD$ of triangle $ABC$ intersects its inscribed circle (with center $O$) at the points $X$ and $Y$. Find the angle $XOY$ if $AC = AB + AD$.
(A Fedotov)
The base $AD$ of a trapezoid $ABCD$ is twice greater than the base $BC$, and the angle $C$ equals one and a half of the angle $A$. The diagonal $AC$ divides angle $C$ into two angles. Which of them is greater?
(i) $ABC$ is a triangle with a right angle at $A$, and $P$ is a point on the hypotenuse $BC$.
The line $AP$ produced beyond $P$ meets the line through $B$ which is perpendicular to $BC$ at $U$.
Prove that $BU = BA$ if, and only if, $CP = CA$.
(ii) $A$ is a point on the semicircle $CB$, and points $X$ and $Y$ are on the line segment $BC$.
The line $AX$, produced beyond $X$, meets the line through $B$ which is perpendicular to $BC$ at $U$.
Also the line $AY$, produced beyond $Y$, meets the line through $C$ which is perpendicular to $BC$ at $V$.
Given that $BY = BA$ and $CX = CA$, determine the angle $\angle VAU$.
Inside triangle $ABC$, point $P$ is taken so that angles $\angle ARB= \angle BPC = \angle CPA= 120^o$. Lines $BP$ and $CP$ intersect lines $AC$ and $AB$ at points $M$ and $K$. It is known that the quadrilateral $AMPK$ has same areq with the triangle $BCP$. What is the angle $\angle BAC$?
Let $ABC$ be a triangle with incircle $(I)$, tangent to $BC$, $CA$, $AB$ at $D, E, F$ respectively. On the line $DF$, take points $M, P$ such that $CM \parallel AB$, $AP \parallel BC$. On the line $DE$, take points $N$, $Q$ such that $BN \parallel AC$, $AQ \parallel BC$. Denote $X$ as intersection of $PE$, $QF$ and $K$ as the midpoint of $BC$. Prove that if $AX = IK$ then $\angle BAC \le 60^o$.
The feet of the altitudes in the triangle $ABC$ are $A', B', C'$. Find the angles of $A'B'C'$ in terms of the angles $A, B, C$. Show that the largest angle in $A'B'C'$ is at least as big as the largest angle in $ABC$. When is it equal?
Let $ABCD$ be a trapezium with $AC = BC$. Let $H$ be the midpoint of the base $AB$ and let $\ell$ be a line passing through $H$. Let $\ell$ meet $AD$ at $P$ and $BD$ at $Q$. Prove that the angles $ACP$ and $QCB$ are either equal or have a sum of $180^o$.
(I. Sharygin, Moscow)
Let $AB$ be a chord of a circle with centre $O$. Let $C$ be a point on the circle such that $\angle ABC =30^o$ and $O$ lies inside the triangle $ABC$. Let $D$ be a point on $AB$ such that $\angle DCO = \angle OCB = 20^o$. Find the measure of $\angle CDO$ in degrees.
Let $ABC$ be a triangle with $\angle A=105^o$ and $\angle C=\frac{1}{4} \angle B$.
a) Find the angles $\angle B$ and $\angle C$
b) Let $O$ be the center of the circumscribed circle of the triangle $ABC$ and let $BD$ be a diameter of that circle. Prove that the distance of point $C$ from the line $BD$ is equal to $\frac{BD}{4}$.
In triangle $ABC$, the length of the angle bisector $AD$ is $\sqrt{BD \cdot CD}$. Find the angles of the triangle $ABC$, if $\angle ADB = 45^o$.
One side of a triangle is equal to one third of the sum of the other two. Prove that the angle opposite the first side is the smallest angle of the triangle.
(AK Tolpygo)
In the right trapezoid $ABCD$ with $AB \parallel CD, \angle B = 90^o$ and $AB = 2DC$.
At points $A$ and $D$ there is therefore a part of the plane $(ABC)$ perpendicular to the plane of the trapezoid, on which the points $N$ and $P$ are taken, ($AP$ and $PD$ are perpendicular to the plane) such that $DN = a$ and $AP = \frac{a}{2}$ . Knowing that $M$ is the midpoint of the side $BC$ and the triangle $MNP$ is equilateral, determine:
a) the cosine of the angle between the planes $MNP$ and $ABC$.
b) the distance from $D$ to the plane $MNP$
The point $P$ lies on the edge $AB$ of a quadrilateral $ABCD$.
The angles $BAD, ABC$ and $CPD$ are right, and $AB = BC + AD$.
Show that $BC = BP$ or $AD = BP$.
Let $E$ be the midpoint of the side $CD$ of a square $ABCD$. Consider the point $M$ inside the square such that $\angle MAB = \angle MBC = \angle BME = x$. Find the angle $x$.
Let $ABC$ be a triangle with incenter $I$. Let $M_b$ and $M_a$ be the midpoints of $AC$ and $BC$, respectively. Let $B'$ be the point of intersection of lines $M_bI$ and $BC$, and let $A'$ be the point of intersection of lines $M_aI$ and $AC$.
If triangles $ABC$ and $A'B'C$ have the same area, what are the possible values of $\angle ACB$?
Consider the number of triangles where the side lengths $a,b,c$ satisfy $(a + b + c) (a + b -c) = 2b^2$.
Determine the angles in the triangle for which the angle opposite to the side with the length $a$ is as big as possible.
Point $D$ on the side $BC$ of the acute triangle $ABC$ is chosen so that $AD = AC$. Let $P$ and $Q$ be the feet of the perpendiculars from $C$ and $D$ on the side $AB$, respectively. Suppose that $AP^2 + 3BP^2 = AQ^2 + 3BQ^2$. Determine the measure of angle $\angle ABC$.
The figure below shows a regular $ABCDE$ pentagon inscribed in an equilateral triangle $MNP$ . Determine the measure of the angle $CMD$.
[img]http://4.bp.blogspot.com/-LLT7hB7QwiA/Xp9fXOsihLI/AAAAAAAAL14/5lPsjXeKfYwIr5DyRAKRy0TbrX_zx1xHQCK4BGAYYCw/s200/2012%2Bobm%2Bl2.png[/img]
For a triangle $T = ABC$ we take the point $X$ on the side $(AB)$ such that $AX/AB=4/5$, the point $Y$ on the segment $(CX)$ such that $CY = 2YX$ and, if possible, the point $Z$ on the ray ($CA$ such that $\widehat{CXZ} = 180 - \widehat{ABC}$. We denote by $\Sigma$ the set of all triangles $T$ for which
$\widehat{XYZ} = 45$. Prove that all triangles from $\Sigma$ are similar and find the measure of their smallest angle.
Let the circles $C_1, C_2$, and $C_3$ be orthogonal to the circle $C$ and intersect each other inside $C$ forming acute angles of measures $A, B$, and $C$. Show that $A + B +C < \pi.$
Given a cube $ABCDA_1B_1C_1D_1$ with edge $5$. On the edge $BB_1$ of the cube , point $K$ such thath $BK=4$.
a) Construct a cube section with the plane $a$ passing through the points $K$ and $C_1$ parallel to the diagonal $BD_1$.
b) Find the angle between the plane $a$ and the plane $BB_1C_1$.
Let $ABCDEF $ be a regular hexagon. On the line $AF $ mark the point $X$so that $ \angle DCX = 45^o$ . Find the value of the angle $FXE$.
(Vyacheslav Yasinsky)
Three sides of a trapezoid are equal, and a circle with the longer base as a diameter halves the two non-parallel sides. Find the angles of the trapezoid.
Consider a triangle $ABC$. Let $A_1$ be the symmetric point of $A$ with respect to the line $BC$, $B_1$ the symmetric point of $B$ with respect to the line $CA$, and $C_1$ the symmetric point of $C$ with respect to the line $AB$. Determine the possible set of angles of triangle $ABC$ for which $A_1B_1C_1$ is equilateral.