Found problems: 300
Let $ABCD$ be an isosceles trapezoid, with a large base $CD$ and a small base $AB$. Let $M$ be any point on side $AB$ and $(d)$ be the line through $M$ and perpendicular to $AB$. Two rays $Mx$ and $My$ are said to satisfy the condition $(T)$ if they are symmetric about each other through $(d)$ and intersect the two rays $AD$ and $BC$ at $E$ and $F$ respectively. Find the locus of the midpoint of the segment $EF$ when the two rays $Mx$ and $My$ change and satisfy condition $(T)$.
Let $ABC$ a triangle. Let $D$ be a point on the circumcircle of this triangle and let $E , F$ be the feet of the perpendiculars from $A$ on $DB, DC$, respectively. Finally, let $N$ be the midpoint of $EF$. Let $M \ne N$ be the midpoint of the side $BC$ . Prove that the lines $NA$ and $NM$ are perpendicular.
Let $ABC$ be a triangle, $P$ the midpoint of $BC$, and $Q$ a point on segment $CA$ such that $|CQ| = 2|QA|$. Let $S$ be the intersection of $BQ$ and $AP$. Prove that $|AS| = |SP|$.
Let $ABC$ be an acute triangle and $M$ be the midpoint of $AB$. A circle through the points $B$ and $C$ intersects the segments $CM$ and $BM$ at points $P$ and $Q$ respectively. Point $K$ is symmetric to $P$ with respect to point $M$. The circumcircles of $\triangle AKM$ and $\triangle CQM$ intersect for the second time at $X$. The circumcircles of $\triangle AMC$ and $\triangle KMQ$ intersect for the second time at $Y$. The segments $BP$ and $CQ$ intersect at point $T$. Prove that the line $MT$ is tangent to the circumcircle of $\triangle MXY$.
Given that $ABC$ is a triangle, points $A_i, B_i, C_i \hspace{0.15cm} (i \in \{1,2,3\})$ and $O_A, O_B, O_C$ satisfy the following criteria:
a) $ABB_1A_2, BCC_1B_2, CAA_1C_2$ are rectangles not containing any interior points of the triangle $ABC$,
b) $\displaystyle \frac{AB}{BB_1} = \frac{BC}{CC_1} = \frac{CA}{AA_1}$,
c) $AA_1A_3A_2, BB_1B_3B_2, CC_1C_3C_2$ are parallelograms, and
d) $O_A$ is the centroid of rectangle $BCC_1B_2$, $O_B$ is the centroid of rectangle $CAA_1C_2$, and $O_C$ is the centroid of rectangle $ABB_1A_2$.
Prove that $A_3O_A, B_3O_B,$ and $C_3O_C$ concur at a point.
[i]Proposed by Farras Mohammad Hibban Faddila[/i]
A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.
Let $AB$ and $A_1B_1$ be two skew segments, $O$ and $O_1$ their respective midpoints. Prove that $OO_1$ is shorter than a half sum of $AA_1$ and $BB_1$.
In the quadrilateral $ABCD$ point $E$ - the midpoint of the side $AB$, point $F$ - the midpoint of the side $BC$, point $G$ - the midpoint $AD$ . It turned out that the segment $GE$ is perpendicular to $AB$, and the segment $GF$ is perpendicular to the segment $BC$. Find the value of the angle $GCD$, if it is known that $\angle ADC = 70 {} ^ \circ$.
Consider a triangle $ABC$ with $AB < AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. A line starting from $B$ cuts the lines $AO$ and $AH$ at $M$ and $M'$ so that $M'$ is the midpoint of $BM$. Another line starting from $C$ cuts the lines $AH$ and $AO$ at $N$ and $N'$ so that $N'$ is the midpoint of $CN$. Prove that $M, M', N, N'$ are on the same circle.
Let $ABC$ be a triangle with $\angle A<90^o, AB \ne AC$. Denote $H$ the orthocenter of triangle $ABC$, $N$ the midpoint of segment $[AH]$, $M$ the midpoint of segment $[BC]$ and $D$ the intersection point of the angle bisector of $\angle BAC$ with the segment $[MN]$. Prove that $<ADH=90^o$
$E$ and $F$ are the midpoints of the sides $BC$ and $AD$ of the convex quadrilateral $ABCD$. Prove that the segment $EF$ divides the diagonals $AC$ and $BD$ in the same ratio.
In a triangle $ABC, A_1,B_1,C_1$ are the midpoints of segments $BC,CA,AB, A_2,B_2,C_2$ are the midpoints of segments $B_1C_1,C_1A_1,A_1B_1$, and $A_3,B_3,C_3$ are the incenters of triangles $B_1AC_1,C_1BA_1,A_1CB_1$, respectively. Show that the lines $A_2A_3,B_2B_3$ and $C_2C_3$ are concurrent.
In a triangle $ABC$, let $D$ and $E$ point in the sides $BC$ and $AC$ respectively. The segments $AD$ and $BE$ intersects in $O$, let $r$ be line (parallel to $AB$) such that $r$ intersects $DE$ in your midpoint, show that the triangle $ABO$ and the quadrilateral $ODCE$ have the same area.
Let $P$ and $Q$ be the midpoints of the sides $BC$ and $CD$, respectively in a rectangle $ABCD$. Let $K$ and $M$ be the intersections of the line $PD$ with the lines $QB$ and $QA$, respectively, and let $N$ be the intersection of the lines $PA$ and $QB$. Let $X$, $Y$ and $Z$ be the midpoints of the segments $AN$, $KN$ and $AM$, respectively. Let $\ell_1$ be the line passing through $X$ and perpendicular to $MK$, $\ell_2$ be the line passing through $Y$ and perpendicular to $AM$ and $\ell_3$ the line passing through $Z$ and perpendicular to $KN$. Prove that the lines $\ell_1$, $\ell_2$ and $\ell_3$ are concurrent.
Let $O$ be a point inside an acute angle with the vertex $A$ and $H, N$ be the feet of the perpendiculars drawn from $O$ onto the sides of the angle. Let point $B$ belong to the bisector of the angle, $K$ be the foot of the perpendicular from $B$ onto either side of the angle. Denote by $P,F$ the midpoints of the segments $AK,HN$ respectively. Known that $ON + OH = BK$, prove that $PF$ is perpendicular to $AB$.
Ya. Konstantinovski
Given an equilateral triangle $ABC$ and a straight line $\ell$, passing through its center. Intersection points of this line with sides $AB$ and $BC$ are reflected wrt to the midpoints of these sides respectively. Prove that the line passing through the resulting points, touches the inscribed circle triangle $ABC$.
A triangle $ABC$ is given. The segment connecting the points where the excircles touch $AB$ and $AC$ meets the bisector of angle $C$ at $X$. The segment connecting the points where the excircles touch $BC$ and $AC$ meets the bisector of angle $A$ at $Y$. Prove that the midpoint of $XY$ is equidistant from $A$ and $C$.
Let $P$ be an interior point of a circle $\Gamma$ centered at $O$ where $P \ne O$. Let $A$ and $B$ be distinct points on $\Gamma$. Lines $AP$ and $BP$ meet $\Gamma$ again at $C$ and $D$, respectively. Let $S$ be any interior point on line segment $PC$. The circumcircle of $\vartriangle ABS$ intersects line segment $PD$ at $T$. The line through $S$ perpendicular to $AC$ intersects $\Gamma$ at $U$ and $V$ . The line through $T$ perpendicular to $BD$ intersects $\Gamma$ at $X$ and $Y$ . Let $M$ and $N$ be the midpoints of $UV$ and $XY$ , respectively. Let $AM$ and $BN$ meet at $Q$. Suppose that $AB$ is not parallel to $CD$. Show that $P, Q$, and $O$ are collinear if and only if $S$ is the midpoint of $PC$.
Let $O$ and $I$ be the centers of the circumscribed and inscribed circle the acute-angled triangle $ABC$, respectively. It is known that line $OI$ is parallel to the side $BC$ of this triangle. Line $MI$, where $M$ is the midpoint of $BC$, intersects the altitude $AH$ at the point $T$. Find the length of the segment $IT$, if the radius of the circle inscribed in the triangle $ABC$ is equal to $r$.
(Grigory Filippovsky)
On the circle $k$, the points $A,B$ are given, while $AB$ is not the diameter of the circle $k$. Point $C$ moves along the long arc $AB$ of circle $k$ so that the triangle $ABC$ is acute. Let $D,E$ be the feet of the altitudes from $A, B$ respectively. Let $F$ be the projection of point $D$ on line $AC$ and $G$ be the projection of point $E$ on line $BC$.
(a) Prove that the lines $AB$ and $FG$ are parallel.
(b) Determine the set of midpoints $S$ of segment $FG$ while along all allowable positions of point $C$.
Let $ABC$ be an acute, non-isosceles triangle with the circumcircle $(O)$. Denote $D, E$ as the midpoints of $AB,AC$ respectively. Two circles $(ABE)$ and $(ACD)$ intersect at $K$ differs from $A$. Suppose that the ray $AK$ intersects $(O)$ at $L$. The line $LB$ meets $(ABE)$ at the second point $M$ and the line $LC$ meets $(ACD)$ at the second point $N$.
a) Prove that $M, K, N$ collinear and $MN$ perpendicular to $OL$.
b) Prove that $K$ is the midpoint of $MN$
In a triangle $ABC$ let $D,E,Z,H,G$ be the midpoints of $BC,AD,BD,ED,EZ$ respectively. Let $I$ be the intersection of $BE,AC$ and let $K$ be the intersection of $HG,AC$. Prove that:
a) $AK=3CK$
b) $HK=3HG$
c) $BE=3EI$
d) $(EGH)=\frac{1}{32}(ABC)$
Notation $(...)$ stands for area of $....$
In a quadrilateral $ABCD$ points $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$, respectively. The line through $M$ and $N$ meets $AB$ and $CD$ at $M'$ and $N'$, respectively. Prove that if $MM' = NN'$, then $AD // BC$.
Given a triangle $ABC$. The circle $\omega_1$ passes through the vertex $B$ and touches the side $AC$ at the point $A$, and the circle $\omega_2$ passes through the vertex $C$ and touches the side $AB$ at the point $A$. The circles $\omega_1$ and $\omega_2$ intersect a second time at the point $D$. The line $AD$ intersects the circumcircle of the triangle $ABC$ at point $E$. Prove that $D$ is the midpoint of $AE$..
Points $A_1,B_1,C_1$ are selected on the sides $BC$,$CA$,$AB$ respectively of an equilateral triangle $ABC$ in such a way that the inradii of the triangles $C_1AB_1$, $A_1BC_1$, $B_1CA_1$ and $A_1B_1C_1$ are equal. Prove that $A_1,B_1,C_1$ are the midpoints of the corresponding sides.