Found problems: 412
In a cyclic quadrilateral $ABCD$ the line passing through the midpoint of $AB$ and the intersection point of the diagonals is perpendicular to $CD$. Prove that either the sides $AB$ and $CD$ are parallel or the diagonals are perpendicular.
Let $ABC$ be an isosceles acute triangle ($AB = BC$). On the side $BC$ we mark a point $P$, such that $\angle PAC = 45^o$, and $Q$ is the point of intersection of the perpendicular bisector of the segment $AP$ with the side $AB$. Prove that $PQ \perp BC$.
Two circle $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$, point $P$ is not on the line $AB$. Line $AP$ intersects again $\Gamma_1$ and $\Gamma_2$ at points $K$ and $L$ respectively, line $BP$ intersects again $\Gamma_1$ and $\Gamma_2$ at points $M$ and $N$ respectively and all the points mentioned so far are different. The centers of the circles circumscribed around the triangles $KMP$ and $LNP$ are $O_1$ and $O_2$ respectively. Prove that $O_1O_2$ is perpendicular to $AB$.
Let $P$ be a convex polygon, so have all interior angles smaller than $180^o$, and let $X$ be a point in the interior of $P$. Prove that $P$ has a side $[AB]$ such that the perpendicular from $X$ to the line $AB$ lies on the side $[AB]$.
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, non isosceles with $I$ is its incenter. Denote $D, E$ as tangent points of $(I)$ on $AB,AC$, respectively. The median segments respect to vertex $A$ of triangles $ABE$ and $ACD$ meet$ (I)$ at$ P,Q,$ respectively. Take points $M, N$ on the line $DE$ such that $AM \parallel BE$ and $AN \parallel C D$ respectively.
a) Prove that $A$ lies on the radical axis of $(MIP)$ and $(NIQ)$.
b) Suppose that the orthocenter $H$ of triangle $ABC$ lies on $(I)$. Prove that there exists a line which is tangent to three circles of center $A, B, C$ and all pass through $H$.
In triangle $ABC$ $\angle B = 60^o, O$ is the circumcenter, and $L$ is the foot of an angle bisector of angle $B$. The circumcirle of triangle $BOL$ meets the circumcircle of $ABC$ at point $D \ne B$. Prove that $BD \perp AC$.
(D. Shvetsov)
Two non intersecting circles with centers $O_1$ and $O_2$ are tangent to line $s$ at points $A_1$ and $A_2$, respectively, and lying on the same side of this line. Line $O_1O_2$ intersects the first circle at $B_1$ and the second at $B_2$. Prove that the lines $A_1B_1$ and $A_2B_2$ are perpendicular to each other.
Let $P$ be a point on the side $AB$ of a square $ABCD$ and $Q$ a point on the side $BC$. Let $H$ be the foot of the perpendicular from $B$ to $PC$. Suppose that $BP = BQ$. Prove that $QH$ is perpendicular to $HD$.
In the acute-angled triangle $ABC$, let $AP$ and $BQ$ be the altitudes, $CM$ be the median . Point $R$ is the midpoint of $CM$. Line $PQ$ intersects line $AB$ at $T$. Prove that $OR \perp TC$, where $O$ is the center of the circumscribed circle of triangle $ABC$.
Let $ABC$ be a an acute triangle. Points $A_1, B_1$ and $C_1$ are respectively the projections of the vertices $A, B$ and $C$ on the opposite sides of the triangle, the point $H$ is the orthocenter of the triangle, and the point $P$ is the middle of the segment $[AH]$. The lines $BH$ and $A_1C_1$, $P B_1$ and $AB$ intersect respectively at the points $M$ and $N$. Prove that the lines $MN$ and $BC$ are perpendicular.
A trapezoid $ABCD$ with perpendicular diagonals $AC$ and $BD$ is inscribed in a circle $k$. Let $k_a$ and $k_c$ respectively be the circles with diameters $AB$ and $CD$. Compute the area of the region which is inside the circle $k$, but outside the circles $k_a$ and $k_c$.
Let $ABCD$ be a parallelogram which is not a rectangle and $E$ be the point in its plane such that $AE \perp AB$ and $CE \perp CB$. Prove that $\angle DEA = \angle CEB$.
In triangle $ABC$, point $I$ is the center of the inscribed circle points, points $I_A$ and $I_C$ are the centers of the excircles, tangent to sides $BC$ and $AB$, respectively. Point $O$ is the center of the circumscribed circle of triangle $II_AI_C$. Prove that $OI \perp AC$
Let $ABCD$ be a square. Let $M$ and $K$ be points on segments $BC$ and $CD$ respectively, such that $MC = KD$. Let $ P$ be the intersection of the segments $MD$ and $BK$. Prove that $AP$ is perpendicular to $MK$.
Given a triangle $ABC$, in which $AB = BC$. Point $O$ is the center of the circumcircle, point $I$ is the center of the incircle. Point $D$ lies on the side $BC$, such that the lines $DI$ and $AB$ parallel. Prove that the lines $DO$ and $CI$ are perpendicular.
(Vyacheslav Yasinsky)
Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.
In the cyclic quadrilateral $ABCD$, the sides $AB, DC$ meet at $Q$, the sides $AD,BC$ meet at $P, M$ is the midpoint of $BD$, If $\angle APQ=90^o$, prove that $PM$ is perpendicular to $AB$.
Let $\Gamma_1$ and $\Gamma_2$ be circles - with respective centres $O_1$ and $O_2$ - that intersect each other in $A$ and $B$. The line $O_1A$ intersects $\Gamma_2$ in $A$ and $C$ and the line $O_2A$ intersects $\Gamma_1$ in $A$ and $D$. The line through $B$ parallel to $AD$ intersects $\Gamma_1$ in $B$ and $E$. Suppose that $O_1A$ is parallel to $DE$. Show that $CD$ is perpendicular to $O_2C$.
The quadrilateral $ABCD$ is inscribed in the circle $o$. Bisectors of angles $DAB$ and $ABC$ intersect at point $P$, and bisectors of angles $BCD$ and $CDA$ intersect in point $Q$. Point $M$ is the center of this arc $BC$ of the circle $o$ which does not contain points $D$ and $A$. Point $N$ is the center of the arc $DA$ of the circle $o$, which does not contain points $B$ and $C$. Prove that the points $P$ and $Q$ lie on the line perpendicular to $MN$.
Suppose a triangle $\vartriangle ABC$ with $\angle C = 90^o$ is given. Let $D$ be the midpoint of $AC$, and let $E$ be the foot of the altitude through $C$ on $BD$. Show that the tangent in $C$ of the circumcircle of $\vartriangle AEC$ is perpendicular to $AB$.
In triangle $ABC, \angle B = 60^o$, $O$ is the circumcenter, and $L$ is the foot of an angle bisector of angle $B$.The circumcirle of triangle $BOL$ meets the circumcircle of $ABC$ at point $D \ne B$. Prove that $BD \perp AC$.
Let $P$ be a $2016$ sided polygon with all its adjacent sides perpendicular to each other, i.e., all its internal angles are either $90^o$ or $270^o$. If the lengths of its sides are odd integers, prove that its area is an even integer.
In the acute triangle $ABC$, let $D$ be the foot of the perpendicular from $A$ to $BC$, let $E$ be the foot of the perpendicular from $D$ to $AC$, and let $F$ be a point on the line segment $DE$. Prove that $AF$ is perpendicular to $BE$ if and only if $FE/FD = BD/DC$
In a circle with radius $R$, there is inscribed a quadrilateral with perpendicular diagonals. From the intersection point of the diagonals, there are perpendiculars drawn to the sides of the quadrilateral.
(a) Prove that the feet of these perpendiculars $P_1,P_2,P_3,P_4$ are vertices of the quadrilateral that is inscribed and circumscribed.
(b) Prove the inequalities $2r_1\le\sqrt2 R_1\le R$ where $R_1$ and $r_1$ are radii respectively of the circumcircle and inscircle to the quadrilateral $P_1P_2P_3P_4$. When does equality hold?
[i]H. Lesov[/i]