Given a point $ P$ inside a triangle $ \triangle ABC$. Let $ D$, $ E$, $ F$ be the orthogonal projections of the point $ P$ on the sides $ BC$, $ CA$, $ AB$, respectively. Let the orthogonal projections of the point $ A$ on the lines $ BP$ and $ CP$ be $ M$ and $ N$, respectively. Prove that the lines $ ME$, $ NF$, $ BC$ are concurrent. [i]Original formulation:[/i] Let $ ABC$ be any triangle and $ P$ any point in its interior. Let $ P_1, P_2$ be the feet of the perpendiculars from $ P$ to the two sides $ AC$ and $ BC.$ Draw $ AP$ and $ BP,$ and from $ C$ drop perpendiculars to $ AP$ and $ BP.$ Let $ Q_1$ and $ Q_2$ be the feet of these perpendiculars. Prove that the lines $ Q_1P_2,Q_2P_1,$ and $ AB$ are concurrent.

Three rays $h_1,h_2,h_3$ emanating from a point $O$ are given, not all in the same plane. Show that if for any three points $A_1,A_2,A_3$ on $h_1,h_2,h_3$ respectively, distinct from $O$, the triangle $A_1A_2A_3$ is acute-angled, then the rays $h_1,h_2,h_3$ are pairwise orthogonal.

Let $OX, OY, OZ$ be mutually orthogonal lines in space. Let $C$ be a fixed point on $OZ$ and $U,V$ variable points on $OX, OY,$ respectively. Find the locus of a point $P$ such that $PU, PV, PC$ are mutually orthogonal.

The diagonals $AC$ and $BD$ of a convex quarilateral $ABCD$ are orthogonal. Let $M,N,R,S$ be the midpoints of the sides $AB,BC,CD$ and $DA$ respectively, and let $W,X,Y,Z$ be the projections of the points $M,N,R$ and $S$ on the lines $CD,DA,AB$ and $BC$, respectively. Prove that the points $M,N,R,S,W,X,Y$ and $Z$ lie on a circle.

In triangle $ABC$ $AD$ and $BE$ are altitudes. Let $L$ be a point on $ED$ such that $ED$ is orthogonal to $BL$. If $LB^2=LD\cdot LE$ prove that triangle $ABC$ is isosceles

Let $ABC$ be an acute triangle with the circumcircle $k$ and incenter $I$. The perpendicular through $I$ in $CI$ intersects segment $[BC]$ in $U$ and $k$ in $V$. In particular $V$ and $A$ are on different sides of $BC$. The parallel line through $U$ to $AI$ intersects $AV$ in $X$. Prove: If $XI$ and $AI$ are perpendicular to each other, then $XI$ intersects segment $[AC]$ in its midpoint $M$. [i](Notation: $[\cdot]$ denotes the line segment.)[/i]

Given the point $O$ in the space and $1979$ straight lines $l_1, l_2, ... , l_{1979}$ containing it. Not a pair of lines is orthogonal. Given a point $A_1$ on $l_1$ that doesn't coincide with $O$. Prove that it is possible to choose the points $A_i$ on $l_i$ ($i = 2, 3, ... , 1979$) in so that $1979$ pairs will be orthogonal: $A_1A_3$ and $l_2$, $A_2A_4$ and $l_3$,$ ...$ , $A_{i-1}A_{i+1}$ and $l_i$,$ ...$ , $A_{1977}A_{1979}$ and $l_{1978}$, $A_{1978}A_1$ and $l_{1979}$, $A_{1979}A_2$ and $l_1$

The diagonals of a convex quadrilateral $ABCD$ are orthogonal and intersect at point $E$. Prove that the reflections of $E$ in the sides of quadrilateral $ABCD$ lie on a circle.

The edge $[AC]$ of the tetrahedron $ABCD$ is orthogonal to $[BC]$, and $[AD]$ is orthogonal to $[BD]$. Prove that the cosine of the angle between lines $(AC)$ and $(BD)$ is less than $|CD|/|AB|$.

Consider a cyclic quadrilateral $ABCD$ and define points $X = AB \cap CD$, $Y = AD \cap BC$, and suppose that there exists a circle with center $Z$ inscribed in $ABCD$. Show that the $Z$ belongs to the circle with diameter $XY$ , which is orthogonal to circumcircle of $ABCD$.

Three circles are pair-wise tangent to each other. Prove that the circle passing through the three tangent points is perpendicular to each of the initial three circles.

Let $ABC$ be an acute triangle with the circumcircle $k$ and incenter $I$. The perpendicular through $I$ in $CI$ intersects segment $[BC]$ in $U$ and $k$ in $V$. In particular $V$ and $A$ are on different sides of $BC$. The parallel line through $U$ to $AI$ intersects $AV$ in $X$. Prove: If $XI$ and $AI$ are perpendicular to each other, then $XI$ intersects segment $[AC]$ in its midpoint $M$. [i](Notation: $[\cdot]$ denotes the line segment.)[/i]