Found problems: 28
Let $AA',BB',CC'$ be parallel lines not lying in the same plane. Denote $U$ the intersection of the planes $A'BC,AB'C,ABC'$ and $V$ the intersection of the planes $AB'C',A'BC',A'B'C$. Show that the line $UV$ is parallel with $AA'$.
A square with side $1$ is intersected by two parallel lines as shown in the figure. Find the sum of the perimeters of the shaded triangles if the distance between the lines is also $1$.
[img]https://cdn.artofproblemsolving.com/attachments/9/e/4e70610b80871325a72e923a0909eff06aebfa.png[/img]
Let $ABC$ be a triangle.
$r$ and $s$ are the angle bisectors of $\angle ABC$ and $\angle BCA$, respectively.
The points $E$ in $r$ and $D$ in $s$ are such that $AD \| BE$ and $AE \| CD$.
The lines $BD$ and $CE$ cut each other at $F$.
$I$ is the incenter of $ABC$.
Show that if $A,F,I$ are collinear, then $AB=AC$.
Let $L_1$ and $L_2$ be two parallel lines and $L_3$ a line perpendicular to $L_1$ and $L_2$ at $H$ and $P$, respectively. Points $Q$ and $R$ lie on $L_1$ such that $QR = PR$ ($Q \ne H$). Let $d$ be the diameter of the circle inscribed in the triangle $PQR$. Point $T$ lies $L_2$ in the same semiplane as $Q$ with respect to line $L_3$ such that $\frac{1}{TH}= \frac{1}{d}- \frac{1}{PH}$ . Let $X$ be the intersection point of $PQ$ and $TH$. Find the locus of the points $X$ as $Q$ varies on $L_1$.
Triangles $ABC$ and $DEF$ have pairwise parallel sides: $EF \| BC, FD \| CA$, and $DE \| AB$. The line $m_A$ is the reflection of $EF$ through $BC$, similarly $m_B$ is the reflection of $FD$ through $CA$, and $m_C$ the reflection of $DE$ through $AB$. Assume that the lines $m_A, m_B$, and $m_C$ meet in a common point. What is the ratio between the areas of triangles $ABC$ and $DEF$?
In triangle $ABC$, let the circumcenter, incenter, and orthocenter be $O$, $I$, and $H$ respectively. Segments $AO$, $AI$, and $AH$ intersect the circumcircle of triangle $ABC$ at $D$, $E$, and $F$. $CD$ intersects $AE$ at $M$ and $CE$ intersects $AF$ at $N$. Prove that $MN$ is parallel to $BC$.
In the plane, a set of points $ M $ is given with the following properties:
1. The points of the set $ M $ do not lie on one straight line,
2. If the points $ A, B, C$, and $D$ are vertices of a parallelogram and $ A, B, C \in M $, then $ D \in M $,
3. If $ A, B \in M $, then $ AB \geq 1 $.
Prove that there exist two families of parallel lines such that $ M $ is the set of all intersection points of the lines of the first family with the lines of the second family.
In triangle $ABC$ $AA_0$ and $BB_0$ are medians, $AA_1$ and $BB_1$ are altitudes. The circumcircles of triangles $CA_0B_0$ and $CA_1B_1$ meet again in point $M_c$. Points $M_a, M_b$ are defined similarly. Prove that points $M_a, M_b, M_c$ are collinear and lines $AM_a, BM_b, CM_c$ are parallel.
Three parallel lines $d_a, d_b, d_c$ pass through the vertex of triangle $ABC$. The reflections of $d_a, d_b, d_c$ in $BC, CA, AB$ respectively form triangle $XYZ$. Find the locus of incenters of such triangles.
(C.Pohoata)
In the inscribed quadrilateral $ABCD$, $P$ is the intersection point of diagonals and $M$ is the midpoint of arc $AB$. Prove that line $MP$ passes through the midpoint of segment $CD$, if and only if lines $AB, CD$ are parallel.
Let \( M \) and \( N \) be the midpoints of sides \( BC \) and \( AC \), respectively, in an acute-angled triangle \( ABC \). Suppose there exists a point \( P \) on the line segment \( AM \) such that \( \angle NPC = \angle MPC \). Let \( D \) be the intersection point of the line \( NP \) and the line parallel to \( CP \) passing through \( B \). Prove that \( AD = AB \).
Proposed by Soroush Behroozifar
Let $ABC$ be a triangle with $AB \neq AC$. The incircle of $ABC$ touches the sides $BC$, $CA$, $AB$ at $X$, $Y$, $Z$ respectively. The line through $Z$ and $Y$ intersects $BC$ extended in $X^\prime$. The lines through $B$ that are parallel to $AX$ and $AC$ intersect $AX^\prime$ in $K$ and $L$ respectively. Prove that $AK = KL$.
Let $ ABC$ be a triangle with circum-circle $ \Gamma$. Let $ M$ be a point in the interior of triangle $ ABC$ which is also on the bisector of $ \angle A$. Let $ AM, BM, CM$ meet $ \Gamma$ in $ A_{1}, B_{1}, C_{1}$ respectively. Suppose $ P$ is the point of intersection of $ A_{1}C_{1}$ with $ AB$; and $ Q$ is the point of intersection of $ A_{1}B_{1}$ with $ AC$. Prove that $ PQ$ is parallel to $ BC$.
Let the incircle of triangle $ABC$ touch sides $BC, CA$ and $AB$ at the points $D, E$ and $F$ respectively and let $I$ be the center of that circle. Furthermore, let $P$ be the foot of the perpendicular from point $I$ to line $AD$ and let $M$ be the midpoint of $DE$. If $N$ is the intersection point of $PM$ and $AC$, prove that $DN \parallel EF$.
The lines $\ell_1$ and \ell_2 are parallel. The points $A_1,A_2, ...,A_7$ are on $\ell_1$ and the points $B_1,B_2,...,B_8$ are on $\ell_2$. The points are arranged in such a way that the number of internal intersections among the line segments is maximized (example Figure).
The [b]greatest number[/b] of intersection points is
[img]https://cdn.artofproblemsolving.com/attachments/4/9/92153dce5a48fcba0f5175d67e0750b7980e84.png[/img]
A. $580$ B. $585$ C. $588$ D. $590$ E. $593$
Let $ABC$ be a triangle. Choose a point $D$ in its interior. Let $\omega_1$ be a circle passing through $B$ and $D$ and $\omega_2$ be a circle passing through $C$ and $D$ so that the other point of intersection of the two circles lies on $AD$. Let $\omega_1$ and $\omega_2$ intersect side $BC$ at $E$ and $F$, respectively. Denote by $X$ the intersection of $DF$, $AB$ and $Y$ the intersection of $DE, AC$. Show that $XY \parallel BC$.
Let a point $D$ lie on the median $AM$ of a triangle $ABC$. The tangents to the circumcircle of triangle $BDC$ at points $B$ and $C$ meet at point $K$. Prove that $DD'$ is parallel to $AK$, where $D'$ is isogonally conjugated to $D$ with respect to $ABC$.
Given a convex quadrilateral $ABCD$ such that $m(\widehat{ABC})=m(\widehat{BCD})$. The lines $AD$ and $BC$ intersect at a point $P$ and the line passing through $P$ which is parallel to $AB$, intersects $BD$ at $T$. Prove that
$$m(\widehat{ACB})=m(\widehat{PCT})$$
Let $ABCD$ be a trapezoid, where $AD\parallel BC$, $BC<AD$, and $AB\cap DC=T$. A circle $k_1$ is inscribed in $\Delta BCT$ and a circle $k_2$ is an excircle for $\Delta ADT$ which is tangent to $AD$ (opposite to $T$). Prove that the tangent line to $k_1$ through $D$, different than $DC$, is parallel to the tangent line to $k_2$ through $B$, different than $BA$.
Let $ABC$ be an acute triangle and $D$ be the foot of the altitude from $A$ onto $BC$. A semicircle with diameter $BC$ intersects segments $AB, AC$ and $AD$ in the points $F, E$ and $X$, respectively. The circumcircles of the triangles $DEX$ and $DXF$ intersect $BC$ in $L$ and $N$, respectively, other than $D$. Prove that $BN = LC$.
The incircle of triangle $ABC$ touches $BC$, $CA$, $AB$ at points $A_1$, $B_1$, $C_1$, respectively. The perpendicular from the incenter $I$ to the median from vertex $C$ meets the line $A_1B_1$ in point $K$. Prove that $CK$ is parallel to $AB$.
Let $ABCD$ be a trapezoid with $AD \parallel BC$ and $\angle BCD < \angle ABC < 90^\circ$. Let $E$ be the intersection point of the diagonals $AC$ and $BD$. The circumcircle $\omega$ of $\triangle BEC$ intersects the segment $CD$ at $X$. The lines $AX$ and $BC$ intersect at $Y$, while the lines $BX$ and $AD$ intersect at $Z$. Prove that the line $EZ$ is tangent to $\omega$ iff the line $BE$ is tangent to the circumcircle of $\triangle BXY$.
Let $ABCD$ be a quadrilateral (with non-perpendicular diagonals).
The perpendicular from $A$ to $BC$ meets $CD$ at $K$.
The perpendicular from $A$ to $CD$ meets $BC$ at $L$.
The perpendicular from $C$ to $AB$ meets $AD$ at $M$.
The perpendicular from $C$ to $AD$ meets $AB$ at $N$.
1. Prove that $KL$ is parallel to $MN$.
2. Prove that $KLMN$ is a parallelogram if $ABCD$ is cyclic.
A square with side $1$ is intersected by two parallel lines as shown in the figure. Find the sum of the perimeters of the shaded triangles if the distance between the lines is also $1$.
[img]https://cdn.artofproblemsolving.com/attachments/9/e/4e70610b80871325a72e923a0909eff06aebfa.png[/img]
A plane is divided into $N$ regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that $N > 1999$?