In an acute triangle $\triangle ABC, \angle A > \angle B$. Let the midpoint of $AB$ be $D$, and let the foot of the perpendicular from $A$ to $BC$ be $E$, and $B$ from $CA$ be $F$. Let the circumcenter of $\triangle DEF$ be $O$. A point $J$ on segment $BE$ satisfies $\angle ODC = \angle EAJ$. Prove that $AJ \cap DC$ lies on the circumcircle of $\triangle BDE$.

In the trapezoid $ABCD, AB = BC = CD, CH$ is the altitude. Prove that the perpendicular from $H$ on $AC$ passes through the midpoint of $BD$.

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 $E$ be the midpoint of the side $AB$ of the parallelogram $ABCD$. Let $F$ be the projection of $B$ on $AC$. Prove that the triangle $ABF$ is isosceles

Let $AD, BE, CF$ be segments whose midpoints are on the same line $\ell$. The points $X, Y, Z$ lie on the lines $EF, FD, DE$ respectively such that $AX \parallel BY \parallel CZ \parallel \ell$. Prove that $X, Y, Z$ are collinear. Tran Quang Hung, High School of Natural Sciences, Hanoi National University

Let $P,Q,R$ be points on the same side of a line $g$ in the plane. Let $M$ and $N$ be the feet of the perpendiculars from $P$ and $Q$ to $g$ respectively. Point $S$ lies between the lines $PM$ and $QN$ and satisfies and satisfies $PM = PS$ and $QN = QS$. The perpendicular bisectors of $SM$ and $SN$ meet in a point $R$. If the line $RS$ intersects the circumcircle of triangle $PQR$ again at $T$, prove that $S$ is the midpoint of $RT$.

Three points $K, L$ and $M$ are given in the plane. It is known that they are the midpoints of three successive sides of an erased quadrilateral and that these three sides have the same length. Reconstruct the quadrilateral.

Let $K,L, M$ be the midpoints of $BC,CA,AB$ repectively on a given triangle $ABC$. Let $\Gamma$ be a circle passing through $B$ and tangent to the circumcircle of $KLM$, say at $X$. Suppose that $LX$ and $BC$ meet at $\Gamma$ . Show that $CX$ is perpendicular to $AB$.

The incircle of $\vartriangle ABC$ touches the sides $BC,CA,AB$ at $D,E,F$ respectively. A circle through $A$ and $B$ encloses $\vartriangle ABC$ and intersects the line $DE$ at points $P$ and $Q$. Prove that the midpoint of $AB$ lies on the circumircle of $\vartriangle PQF$.

Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $BE = OF$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.

Inside the quadrangle, a point is taken and connected with the midpoint of all sides. Areas of the three out of four formed quadrangles are $S_1, S_2, S_3$. Find the area of the fourth quadrangle.

The touching point of the excircle with the side of a triangle and the base of the altitude to this side are symmetric wrt the base of the corresponding bisector. Prove that this side is equal to one third of the perimeter.

The chords $AB$ and $CD$ of a circle intersect at $M$, which is the midpoint of the chord $PQ$. The points $X$ and $Y$ are the intersections of the segments $AD$ and $PQ$, respectively, and $BC$ and $PQ$, respectively. Show that $M$ is the midpoint of $XY$.

In the acute triangle $ABC$, the midpoint of side $BC$ is called $M$. Point $X$ lies on the angle bisector of $\angle AMB$ such that $\angle BXM = 90^o$. Point $Y$ lies on the angle bisector of $\angle AMC$ such that $\angle CYM = 90^o$. Line segments $AM$ and $XY$ intersect in point $Z$. Prove that $Z$ is the midpoint of $XY$ . [asy] import geometry; unitsize (1.2 cm); pair A, B, C, M, X, Y, Z; A = (0,0); B = (4,1.5); C = (0.5,3); M = (B + C)/2; X = extension(M, incenter(A,B,M), B, B + rotate(90)*(incenter(A,B,M) - M)); Y = extension(M, incenter(A,C,M), C, C + rotate(90)*(incenter(A,C,M) - M)); Z = extension(A,M,X,Y); draw(A--B--C--cycle); draw(A--M); draw(M--interp(M,X,2)); draw(M--interp(M,Y,2)); draw(B--X, dotted); draw(C--Y, dotted); draw(X--Y); dot("$A$", A, SW); dot("$B$", B, E); dot("$C$", C, N); dot("$M$", M, NE); dot("$X$", X, NW); dot("$Y$", Y, NE); dot("$Z$", Z, S); [/asy]

Let $M,N$ be the midpoints of the sides $AD,BC$ respectively of the convex quadrilateral $ABCD$, $K=AN \cap BM$, $L=CM \cap DN$. Find the smallest possible $c\in R$ such that $S(MKNL)<c \cdot S(ABCD)$ for any convex quadrilateral $ABCD$. I. Voronovich

Given is a square with side length $a$. A distance $PQ$ of length $p$, where $p < a$, moves so that its end points are always on the sides of the square. What is the geometric locus of the midpoints of the segments $PQ$?

We consider two circles $k_1(M_1, r_1)$ and $k_2(M_2, r_2)$ with $z = M_1M_2 > r_1+r_2$ and a common outer tangent with the tangent points $P_1$ and $P2$ (that is, they lie on the same side of the connecting line $M_1M_2$). We now change the radii so that their sum is $r_1+r_2 = c$ remains constant. What set of points does the midpoint of the tangent segment $P_1P_2$ run through, when $r_1$ varies from $0$ to $c$?

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]

Points $K$ and $L$ are on side $AB$ of triangle $ABC$ such that $KL=BC$ and $AK=LB$. Let $M$ be a midpoint of $AC$. Prove that $\angle KML = 90^{\circ}$

Let $ABCD$ be a convex quadrilateral and $A',B'C',D'$ be the midpoints of $AB,BC,CD,DA$, respectively. Let $a,b,c,d$ denote the areas of quadrilaterals into which lines $A'C'$ and $B'D'$ divide the quadrilateral $ABCD$ (where a corresponds to vertex $A$ etc.). Prove that $a+c = b+d$.

Let $ABC$ be a triangle, and let $P$ be a point inside it such that $\angle PAC = \angle PBC$. The perpendiculars from $P$ to $BC$ and $CA$ meet these lines at $L$ and $M$, respectively, and $D$ is the midpoint of $AB$. Prove that $DL = DM.$

Let $AA_1$ and $CC_1$ be the altitudes of the acute-angled triangle $ABC$. Let $K,L$ and $M$ be the midpoints of the sides $AB,BC$ and $CA$ respectively. Prove that if $\angle C_1MA_1 =\angle ABC$, then $C_1 K = A_1L$.

In the triangle $ABC$, ${{A}_{1}}$ and ${{C}_{1}} $ are the midpoints of sides $BC $ and $AB$ respectively. Point $P$ lies inside the triangle. Let $\angle BP {{C}_{1}} = \angle PCA$. Prove that $\angle BP {{A}_{1}} = \angle PAC $.

Given a triangle $ABC$, points $K,L,M$ are the midpoints of the sides $BC,CA,AB$, and points $X,Y,Z$ are the midpoints of the arcs $BC,CA,AB$ of the circumcircle not containing $A,B,C$ respectively. If $R$ denotes the circumradius and $r$ the inradius of the triangle, show that $r+KX+LY+MZ=2R$.

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.