Found problems: 412
2016 Switzerland Team Selection Test, Problem 8
Let $ABC$ be a triangle with $AB \neq AC$ and let $M$ be the middle of $BC$. The bisector of $\angle BAC$ intersects the line $BC$ in $Q$. Let $H$ be the foot of $A$ on $BC$. The perpendicular to $AQ$ passing through $A$ intersects the line $BC$ in $S$. Show that $MH \times QS=AB \times AC$.
2011 Indonesia TST, 3
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?
2018 Saudi Arabia GMO TST, 3
Let $C$ be a point lies outside the circle $(O)$ and $CS, CT$ are tangent lines of $(O)$. Take two points $A, B$ on $(O)$ with $M$ is the midpoint of the minor arc $AB$ such that $A, B, M$ differ from $S, T$. Suppose that $MS, MT$ cut line $AB$ at $E, F$. Take $X \in OS$ and $Y \in OT$ such that $EX, FY$ are perpendicular to $AB$. Prove that $X Y$ and $C M$ are perpendicular.
2016 Singapore Senior Math Olympiad, 4
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.
2005 Sharygin Geometry Olympiad, 1
The chords $AC$ and $BD$ of the circle intersect at point $P$. The perpendiculars to $AC$ and $BD$ at points $C$ and $D$, respectively, intersect at point $Q$. Prove that the lines $AB$ and $PQ$ are perpendicular.
1987 Tournament Of Towns, (160) 4
From point $M$ in triangle $ABC$ perpendiculars are dropped to each altitude. It can be shown that each of the line segments of altitudes, measured between the vertex and the foot of the perpendicular drawn to it, are of equal length. Prove that these lengths are each equal to the diameter of the circle inscribed in the triangle.
2015 Hanoi Open Mathematics Competitions, 11
Given a convex quadrilateral $ABCD$. Let $O$ be the intersection point of diagonals $AC$ and $BD$ and let $I , K , H$ be feet of perpendiculars from $B , O , C$ to $AD$, respectively. Prove that $AD \times BI \times CH \le AC \times BD \times OK$.
2021 Dutch Mathematical Olympiad, 4
In triangle $ABC$ we have $\angle ACB = 90^o$. The point $M$ is the midpoint of $AB$. The line through $M$ parallel to $BC$ intersects $AC$ in $D$. The midpoint of line segment $CD$ is $E$. The lines $BD$ and $CM$ are perpendicular.
(a) Prove that triangles $CME$ and $ABD$ are similar.
(b) Prove that $EM$ and $AB$ are perpendicular.
[asy]
unitsize(1 cm);
pair A, B, C, D, E, M;
A = (0,0);
B = (4,0);
C = (2.6,2);
M = (A + B)/2;
D = (A + C)/2;
E = (C + D)/2;
draw(A--B--C--cycle);
draw(C--M--D--B);
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, N);
dot("$D$", D, NW);
dot("$E$", E, NW);
dot("$M$", M, S);
[/asy]
[i]Be aware: the figure is not drawn to scale.[/i]
2021 Czech-Polish-Slovak Junior Match, 2
An acute triangle $ABC$ is given. Let us denote by $D$ and $E$ the orthogonal projections, respectively of points $ B$ and $C$ on the bisector of the external angle $BAC$. Let $F$ be the point of intersection of the lines $BE$ and $CD$. Show that the lines $AF$ and $DE$ are perpendicular.
2024 Greece Junior Math Olympiad, 2
Consider an acute triangle $ABC$ and it's circumcircle $\omega$. With center $A$, we construct a circle $\gamma$ that intersects arc $AB$ of circle $\omega$ , that doesn't contain $C$, at point $D$ and arc $AC$ , that doesn't contain $B$, at point $E$. Suppose that the intersection point $K$ of lines $BE$ and $CD$ lies on circle $\gamma$. Prove that line $AK$ is perpendicular on line $BC$.
2016 Bundeswettbewerb Mathematik, 3
Let $A,B,C$ and $D$ be points on a circle in this order. The chords $AC$ and $BD$ intersect in point $P$. The perpendicular to $AC$ through C and the perpendicular to $BD$ through $D$ intersect in point $Q$.
Prove that the lines $AB$ and $PQ$ are perpendicular.
2019 Junior Balkan Team Selection Tests - Romania, 3
A circle with center $O$ is internally tangent to two circles inside it at points $S$ and $T$. Suppose the two circles inside intersect at $M$ and $N$ with $N$ closer to $ST$. Show that $OM$ and $MN$ are perpendicular if and only if $S,N, T$ are collinear.
XMO (China) 2-15 - geometry, 12.1
As shown in the figure, it is known that the quadrilateral $ABCD$ satisfies $\angle ADB = \angle ACB = 90^o$. Suppose $AC$ and $BD$ intersect at point $P$, point $R$ lies on $CD$ and $RP \perp AB$. $M$ and $N$ are the midpoints of $AB$ and $CD$ respectively. Point $K$ is a point on the extension line of $NM$, the circumscribed circles of $\vartriangle DKC$ and $\vartriangle AKB$ intersect at point $S$. Prove that $KS \perp SR$.
[img]https://cdn.artofproblemsolving.com/attachments/5/d/fc0a391f8ebcdee792e9b226cbf55a058251a1.png[/img]
2008 District Olympiad, 2
Consider the square $ABCD$ and $E \in (AB)$. The diagonal $AC$ intersects the segment $[DE]$ at point $P$. The perpendicular taken from point $P$ on $DE$ intersects the side $BC$ at point $F$. Prove that $EF = AE + FC$.
Durer Math Competition CD 1st Round - geometry, 2016.D+3
Let $M$ be the intersection point of the diagonals of the convex quadrilateral $ABCD$. Let $P$ and $Q$ be the centroids of triangles $AMD$ and $BMC$ respectively. Let $R$ and $S$ are the orthocenters of triangles $AMB$ and $CMD$. Prove that the lines $P Q$ and $RS$ are perpendicular to each other.
2005 Sharygin Geometry Olympiad, 9
Let $O$ be the center of a regular triangle $ABC$. From an arbitrary point $P$ of the plane, the perpendiculars were drawn on the sides of the triangle. Let $M$ denote the intersection point of the medians of the triangle , having vertices the feet of the perpendiculars. Prove that $M$ is the midpoint of the segment $PO$.
1989 Mexico National Olympiad, 1
In a triangle $ABC$ the area is $18$, the length $AB$ is $5$, and the medians from $A$ and $B$ are orthogonal. Find the lengths of the sides $BC,AC$.
2017 Saudi Arabia BMO TST, 3
Let $ABCD$ be a cyclic quadrilateral and triangles $ACD, BCD$ are acute. Suppose that the lines $AB$ and $CD$ meet at $S$. Denote by $E$ the intersection of $AC, BD$. The circles $(ADE)$ and $(BC E)$ meet again at $F$.
a) Prove that $SF \perp EF.$
b) The point $G$ is taken out side of the quadrilateral $ABCD$ such that triangle $GAB$ and $FDC$ are similar. Prove that $GA+ FB = GB + FA$
2000 Czech and Slovak Match, 5
Let $ABCD$ be an isosceles trapezoid with bases $AB$ and $CD$. The incircle of the triangle $BCD$ touches $CD$ at $E$. Point $F$ is chosen on the bisector of the angle $DAC$ such that the lines $EF$ and $CD$ are perpendicular. The circumcircle of the triangle $ACF$ intersects the line $CD$ again at $G$. Prove that the triangle $AFG$ is isosceles.
Durer Math Competition CD Finals - geometry, 2015.D1
From all three vertices of triangle $ABC$, we set perpendiculars to the exterior and interior of the other vertices angle bisectors. Prove that the sum of the squares of the segments thus obtained is exactly $2 (a^2 + b^2 + c^2)$, where $a, b$, and $c$ denote the lengths of the sides of the triangle.
Kyiv City MO Seniors 2003+ geometry, 2015.11.4.1
On the bisector of the angle $ BAC $ of the triangle $ ABC $ we choose the points $ {{B} _ {1}}, \, \, {{C} _ {1}} $ for which $ B {{B} _ {1 }}\perp AB $, $ C {{C} _ {1}} \perp AC $. The point $ M $ is the midpoint of the segment $ {{B} _ {1}} {{C} _ {1}} $. Prove that $ MB = MC $.
2012 District Olympiad, 4
Consider a tetrahedron $ABCD$ in which $AD \perp BC$ and $AC \perp BD$. We denote by $E$ and $F$ the projections of point $B$ on the lines $AD$ and $AC$, respectively. If $M$ and $N$ are the midpoints of the segments $[AB]$ and $[CD]$, respectively, show that $MN \perp EF$
1998 All-Russian Olympiad Regional Round, 11.2
Circle $S$ with center $O$ and circle $S'$ intersect at points $A$ and $B$. Point $C$ is taken on the arc of a circle $S$ lying inside $S'$. Denote the intersection points of $AC$ and $BC$ with $S'$, other than $A$ and $B$, as $E$ and $D$, respectively. Prove that lines $DE$ and $OC$ are perpendicular.
2011 QEDMO 9th, 5
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 $[AB]$.
2008 District Olympiad, 3
Let $ABCDA' B' C' D '$ be a cube , $M$ the foot of the perpendicular from $A$ on the plane $(A'CD)$, $N$ the foot of the perpendicular from $B$ on the diagonal $A'C$ and $P$ is symmetric of the point $D$ with respect to $C$. Show that the points $M, N, P$ are collinear.