Found problems: 478
Let $ P$ be a point inside a triangle $ ABC$ such that
\[ \angle APB \minus{} \angle ACB \equal{} \angle APC \minus{} \angle ABC.
\]
Let $ D$, $ E$ be the incenters of triangles $ APB$, $ APC$, respectively. Show that the lines $ AP$, $ BD$, $ CE$ meet at a point.
On the plane are $n$ paper disks of radius $1$ whose boundaries all pass through a certain point, which lies inside the region covered by the disks. Find the perimeter of this region.
(P Kozhevnikov)
A quadrilateral is circumscribed around a circle. Prove that the straight lines connecting neighboring tangent points either meet on the extension of a diagonal of the quadrilateral or are parallel to it.
$H$ is the intersection point of the heights $AA'$ and $BB'$ of the acute-angled triangle $ABC$. A straight line, perpendicular to $AB$, intersects these heights at points $D$ and $E$, and side $AB$ at point $P$. Prove that the orthocenter of the triangle $DEH$ lies on segment $CP$.
On the side $AC$ of the triangle $ABC$ in the external side is constructed the parallelogram $ACDE$ . Let $O$ be the intersection point of its diagonals, $N$ and $K$ be midpoints of BC and BA respectively. Prove that lines $DK, EN$ and $BO$ intersect at one point.
In a triangle $ABC$ points $A_1,B_1,C_1$ are the midpoints of $BC,CA,AB$ respectively,and $A_2,B_2,C_2$ are the midpoints of the altitudes from $A,B,C$ respectively. Show that the lines $A_1A_2,B_1B_2,C_1,C_2$ are concurrent.
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
The quadrilateral $ABCD$ is inscribed in the circle $\omega$. Lines $AD$ and $BC$ intersect at point $E$. Points $M$ and $N$ are selected on segments $AD$ and $BC$, respectively, so that $AM: MD = BN: NC$. The circumscribed circle of the triangle $EMN$ intersects the circle $\omega$ at points $X$ and $Y$. Prove that the lines $AB, CD$ and $XY$ intersect at the same point or are parallel.
In the acute-angled triangle $ ABC $, the sides $ AB $ and $BC$ have different lengths, and the extension of the median $ BM $ intersects the circumscribed circle at the point $ N $. On this circle we note such a point $ D $ that $ \angle BDH = 90 {} ^ \circ $, where $ H $ is the point of intersection of the altitudes of the triangle $ ABC $. The point $K$ is chosen so that $ ANCK $ is a parallelogram. Prove that the lines $ AC $, $ KH $ and $ BD $ intersect at one point.
(Igor Nagel)
Let $ABC$ be a triangle with $D$ the midpoint of side $AB, E$ the midpoint of side $BC$, and $F$ the midpoint of side $AC$. Let $k_1$ be the circle passing through points $A, D$, and $F$, let $k_2$ be the circle passing through points $B, E$, and $D$, and let $k_3$ be the circle passing through $C, F$, and $E$. Prove that circles $k_1, k_2$, and $k_3$ intersect in a point.
Six different points $A, B, C, D, E, F$ are marked on the plane, no four of them lie on one circle and no two segments with ends at these points lie on parallel lines. Let $P, Q,R$ be the points of intersection of the perpendicular bisectors to pairs of segments $(AD, BE)$, $(BE, CF)$ ,$(CF, DA)$ respectively, and $P', Q' ,R'$ are points the intersection of the perpendicular bisectors to the pairs of segments $(AE, BD)$, $(BF, CE)$ , $(CA, DF)$ respectively. Show that $P \ne P', Q \ne Q', R \ne R'$, and prove that the lines $PP', QQ'$ and $RR'$ intersect at one point or are parallel.
Let $ABC$ be a non-isosceles triangle. The altitude from $A$, the bisector from $B$ and the median from $C$ concur at point $K$.
a) Which of the sidelengths of the triangle is medial (intermediate in length)?
b) Which of the lengths of segments $AK, BK, CK$ is medial (intermediate in length)?
Let $ABCDEF$ be a convex hexagon inscribed in a circle . Prove that the diagonals $AD, BE$ and $CF$ intersect at one point if and only if $$\frac{AB}{BC} \cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1$$
An equilateral triangle $ABC$ is inscribed in a circle $\Omega$ and has incircle $\omega$. Points $P$ and $Q$ are in segments $AC$ and $AB$, respectively, such that $PQ$ is tangent to $\omega$. The circle $\Omega_B$ has center $P$ and radius $PB$ and the circle $\Omega_C$ is defined similarly. Prove that $\Omega$, $\Omega_B$ and $\Omega_C$ have a common point.
Given an isosceles triangle $ABC$ with base $BC$. Inside the side $BC$ is given a point $D$. Let $E, F$ be respectively points on the sides $AB, AC$ that $|\angle BED | = |\angle DF C| > 90^o$ . Prove that the circles circumscribed around the triangles $ABF$ and $AEC$ intersect on the line $AD$ at a point different from point $A$.
(Patrik Bak, Michal Rolínek)
Pentagon $ABCDE$ is given in the plane. Let the perpendicular from $A$ to line $CD$ be $F$, the perpendicular from $B$ to $DE$ be $G$, from $C$ to $EA$ be $H$, from $D$ to $AB$ be $I$,and from $E$ to $BC$ be $J$. Given that lines $AF,BG,CH$, and $DI$ concur, show that they also concur with line $EJ$.
In a triangle $ABC, O$ is the center of the circumscribed circle. Line $a$ passes through the midpoint of the altitude of the triangle from the vertex $A$ and is parallel to $OA$. Similarly, the straight lines $b$ and $c$ are defined. Prove that these three lines intersect at one point.
Let $ABCD$ be a tangential quadrilateral with $BC> BA$. The point $P$ is on the segment $BC$, such that $BP = BA$ . Show that the bisector of $\angle BCD$, the perpendicular on line $BC$ through $P$ and the perpendicular on $BD$ through $A$, intersect at one point.
In a triangle $\triangle ABC$ with incenter $I$, the incircle meets lines $BC, CA, AB$ at $D, E, F$ respectively.
Define the circumcenter of $\triangle IAB$ and $\triangle IAC$ $O_1$ and $O_2$ respectively.
Let the two intersections of the circumcircle of $\triangle ABC$ and line $EF$ be $P, Q$.
Prove that the circumcenter of $\triangle DPQ$ lies on the line $O_1O_2$.
Let $A_1A_2A_3A_4$ be a tetrahedron. For any permutation $(i, j,k,h)$ of $1,2,3,4$ denote:
- $P_i$ – the orthogonal projection of $A_i$ on $A_jA_kA_h$;
- $B_{ij}$ – the midpoint of the edge $A_iAj$,
- $C_{ij}$ – the midpoint of segment $P_iP_j$
- $\beta_{ij}$– the plane $B_{ij}P_hP_k$
- $\delta_{ij}$ – the plane $B_{ij}P_iP_j$
- $\alpha_{ij}$ – the plane through $C_{ij}$ orthogonal to $A_kA_h$
- $\gamma_{ij}$ – the plane through $C_{ij}$ orthogonal to $A_iA_j$.
Prove that if the points $P_i$ are not in a plane, then the following sets of planes are concurrent:
(a) $\alpha_{ij}$, (b) $\beta_{ij}$, (c) $\gamma_{ij}$, (d) $\delta_{ij}$.
Consider the triangle $ABC$, in which $AB > AC$. Let $P$ and $Q$ be the feet of the perpendiculars dropped from the vertices $B$ and $C$ on the bisector of the angle $BAC$, respectively. On the line $BC$ note point $B$ such that $AD \perp AP.$ Prove that the lines $BQ, PC$ and $AD$ intersect at one point.
Let $A_1A_2A_3$ be a triangle and, for $1 \leq i \leq 3$, let $B_i$ be an interior point of edge opposite $A_i$. Prove that the perpendicular bisectors of $A_iB_i$ for $1 \leq i \leq 3$ are not concurrent.
Consider five points on a circle. For every three of them, we draw the perpendicular from the centroid of the triangle they determine to the line through the remaining two points. Prove that the ten lines thus formed are concurrent.
A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$.
[i]Proposed by Mongolia[/i]
Let $ABCD$ be a trapezoid with $AD=a, AB=2a, BC=3a$ and $\angle A=\angle B =90 ^o$. Let $E,Z$ be the midpoints of the sides $AB ,CD$ respectively and $I$ be the foot of the perpendicular from point $Z$ on $BC$. Prove that :
i) triangle $BDZ$ is isosceles
ii) midpoint $O$ of $EZ$ is the centroid of triangle $BDZ$
iii) lines $AZ$ and $DI$ intersect at a point lying on line $BO$