The midpoint of the side $AB$ in the triangle $ABC$ is called $C'$. A point on the side $BC$ is called $D$, and $E$ is the point of intersection of $AD$ and $CC'$. Assume that $AE/ED = 2$. Show that $D$ is the midpoint of $BC$.

A triangle $ABC$ and a point $P$ inside this triangle are given. Define $D, E$ and $F$ as the midpoints of $AP, BP$ and $CP$, respectively. Furthermore, let $R$ be the intersection of $AE$ and $BD, S$ the intersection of $BF$ and $CE$, and $T$ the intersection of $CD$ and $AF$. Prove that the area of hexagon $DRESFT$ is independent of the position of $P$ inside the triangle. [asy] unitsize(1 cm); pair A, B, C, D, E, F, P, R, S, T; A = (0,0); B = (5,0); C = (1.5,4); P = (2,2); D = (A + P)/2; E = (B + P)/2; F = (C + P)/2; R = extension(A,E,B,D); S = extension(B,F,C,E); T = extension(C,D,A,F); draw(A--B--C--cycle); draw(A--P); draw(B--P); draw(C--P); draw(A--F--B); draw(B--D--C); draw(C--E--A); dot("$A$", A, SW); dot("$B$", B, SE); dot("$C$", C, N); dot("$D$", D, dir(270)); dot("$E$", E, dir(270)); dot("$F$", F, W); dot("$P$", P, dir(270)); dot("$R$", R, dir(270)); dot("$S$", S, SW); dot("$T$", T, SE); [/asy]

Let $M,N,P$ be midpoints of $BC,AC$ and $AB$ of triangle $\triangle ABC$ respectively. $E$ and $F$ are two points on the segment $\overline{BC}$ so that $\angle NEC = \frac{1}{2} \angle AMB$ and $\angle PFB = \frac{1}{2} \angle AMC$. Prove that $AE=AF$. [i]Proposed by Alireza Dadgarnia[/i]

Consider a rectangle $ABCD$ with $AB = 2$ and $BC = \sqrt3$. The point $M$ lies on the side $AD$ so that $MD = 2 AM$ and the point $N$ is the midpoint of the segment $AB$. On the plane of the rectangle rises the perpendicular MP and we choose the point $Q$ on the segment $MP$ such that the measure of the angle between the planes $(MPC)$ and $(NPC)$ shall be $45^o$, and the measure of the angle between the planes $(MPC)$ and $(QNC)$ shall be $60^o$. a) Show that the lines $DN$ and $CM$ are perpendicular. b) Show that the point $Q$ is the midpoint of the segment $MP$.

Let $ABC$ ne a right triangle with $\angle ACB=90^o$. Let $E, F$ be respecitvely the midpoints of the $BC, AC$ and $CD$ be it's altitude. Next, let $P$ be the intersection of the internal angle bisector from $A$ and the line $EF$. Prove that $P$ is the center of the circle inscribed in the triangle $CDE$ .

In a triangle $ABC$ , let $P$ be a point on its circumscribed circle (on the arc $AC$ that does not contain $B$). Let $H,H_1,H_2$ and $H_3$ be the orthocenters of triangles $ABC, BCP, ACP$ and $ABP$, respectively. Let $L = PB \cap AC$ and $J = HH_2 \cap H_1H_3$. If $M$ and $N$ are the midpoints of $JH$ and $LP$, respectively, prove that $MN$ and $JL$ intersect at their midpoint.

Let $ABCD$ be a random tetrahedron. Let $E$ and $F$ be the midpoints of segments $AB$ and $CD$, respectively. If the angle $a$ is between $AD$ and $BC$, determine $cos a$ in terms of $EF, AD$ and $BC$.

Let $ABC$ be a right triangle in $A$ , whose leg measures $1$ cm. The bisector of the angle $BAC$ cuts the hypotenuse in $R$, the perpendicular to $AR$ on $R$ , cuts the side $AB$ at its midpoint. Find the measurement of the side $AB$ .

Given two distinct points $A_1,A_2$ in the plane, determine all possible positions of a point $A_3$ with the following property: There exists an array of (not necessarily distinct) points $P_1,P_2,...,P_n$ for some $n \ge 3$ such that the segments $P_1P_2,P_2P_3,...,P_nP_1$ have equal lengths and their midpoints are $A_1, A_2, A_3, A_1, A_2, A_3, ...$ in this order.

Let $BC$ be a chord of a circle $(O)$ such that $BC$ is not a diameter. Let $AE$ be the diameter perpendicular to $BC$ such that $A$ belongs to the larger arc $BC$ of $(O)$. Let $D$ be a point on the larger arc $BC$ of $(O)$ which is different from $A$. Suppose that $AD$ intersects $BC$ at $S$ and $DE$ intersects $BC$ at $T$. Let $F$ be the midpoint of $ST$ and $I$ be the second intersection point of the circle $(ODF)$ with the line $BC$. 1. Let the line passing through $I$ and parallel to $OD$ intersect $AD$ and $DE$ at $M$ and $N$, respectively. Find the maximum value of the area of the triangle $MDN$ when $D$ moves on the larger arc $BC$ of $(O)$ (such that $D \ne A$). 2. Prove that the perpendicular from $D$ to $ST$ passes through the midpoint of $MN$

The two angles of the squares are adjacent, and the extension of the diagonals of one square intersect the diagonal of another square at point $O$ (see figure). Prove that $O$ is the midpoint of $AB$. [img]https://cdn.artofproblemsolving.com/attachments/7/8/8daaaa55c38e15c4a8ac7492c38707f05475cc.png[/img]

Let $ABC$ be a triangle with $AH$ altitude. The point $K$ is chosen on the segment $AH$ as follows such that $AH =3KH$. Let $O$ be the center of the circle circumscribed around by triangle $ABC, M$ and $N$ be the midpoints of $AC$ and AB respectively. Lines $KO$ and $MN$ intersect at the point $Z$, a perpendicular to $OK$ passing through point $Z$ intersects lines $AC$ and $AB$ at points $X$ and $Y$ respectively. Prove that $\angle XKY =\angle CKB$.

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]

Suppose that every two opposite edges of a tetrahedron are orthogonal. Show that the midpoints of the six edges lie on a sphere.

Let $D, E$, and $F$ be respectively the midpoints of the sides $BC, CA$, and $AB$ of $\vartriangle ABC$. Let $H_a, H_b, H_c$ be the feet of perpendiculars from $A, B, C$ to the opposite sides, respectively. Let $P, Q, R$ be the midpoints of the $H_bH_c, H_cH_a$, and $H_aH_b$ respectively. Prove that $PD, QE$, and $RF$ are concurrent.