Found problems: 412
Let $ABCD$ be a rectangle and the arbitrary points $E\in (CD)$ and $F \in (AD)$. The perpendicular from point $E$ on the line $FB$ intersects the line $BC$ at point $P$ and the perpendicular from point $F$ on the line $EB$ intersects the line $AB$ at point $Q$. Prove that the points $P, D$ and $Q$ are collinear.
Two circles $\omega_1$ and $\omega_2$ are tangent to line $\ell$ at the points $A$ and $B$ respectively. In addition, $\omega_1$ and $\omega_2 $are externally tangent to each other at point $D$. Choose a point $E$ on the smaller arc $BD$ of circle $\omega_2$. Line $DE$ intersects circle $\omega_1$ again at point $C$. Prove that $BE \perp AC$.
(Yurii Biletskyi)
Let $M$ be a point in the interior of triangle $ABC$. Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$. Define $B'$ on $CA$ and $C'$ on $AB$ similarly. Define
\[
p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}.
\]
Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?
Let $ABC$ be an acute triangle such that $\angle BAC = 45^o$. Let $D$ a point on $AB$ such that $CD \perp AB$. Let $P$ be an internal point of the segment $CD$. Prove that $AP\perp BC$ if and only if $|AP| = |BC|$.
Let $ABC$ be an acute triangle. Let $H$ denote its orthocenter and $D, E$ and $F$ the feet of its altitudes from $A, B$ and $C$, respectively. Let the common point of $DF$ and the altitude through $B$ be $P$. The line perpendicular to $BC$ through $P$ intersects $AB$ in $Q$. Furthermore, $EQ$ intersects the altitude through $A$ in $N$. Prove that $N$ is the midpoint of $AH$.
Proposed by Karl Czakler
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]
As shown in the figure, given $\vartriangle ABC$ with $\angle B$, $\angle C$ acute angles, $AD \perp BC$, $DE \perp AC$, $M$ midpoint of $DE$, $AM \perp BE$. Prove that $\vartriangle ABC$ is isosceles.
[img]https://cdn.artofproblemsolving.com/attachments/a/8/f553c33557979f6f7b799935c3bde743edcc3c.png[/img]
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 $RT$.
Let $ABCD$ be a convex quadrilateral with perpendicular diagonals.
If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?
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 $AB$.
Let $ABCD$ be a square and let $P$ be a point on side $AB$. The point $Q$ lies outside the square such that $\angle ABQ = \angle ADP$ and $\angle AQB = 90^{\circ}$. The point $R$ lies on the side $BC$ such that $\angle BAR = \angle ADQ$. Prove that the lines $AR, CQ$ and $DP$ pass through a common point.
Let $ABCD$ be a square, point $E$ be the midpoint of the side $BC$. On the side $AB$ mark a point $F$ such that $FE \perp DE$. Prove that $AF + BE = DF$.
(Ercole Suppa, Italy)