Found problems: 56
2008 Federal Competition For Advanced Students, P1, 4
In a triangle $ABC$ let $E$ be the midpoint of the side $AC$ and $F$ the midpoint of the side $BC$. Let $G$ be the foot of the perpendicular from $C$ to $ AB$. Show that $\vartriangle EFG$ is isosceles if and only if $\vartriangle ABC$ is isosceles.
2014 May Olympiad, 4
Let $ABC$ be a right triangle and isosceles, with $\angle C = 90^o$. Let $M$ be the midpoint of $AB$ and $N$ the midpoint of $AC$. Let $ P$ be such that $MNP$ is an equilateral triangle with $ P$ inside the quadrilateral $MBCN$. Calculate the measure of $\angle CAP$
1995 Tuymaada Olympiad, 5
A set consisting of $n$ points of a plane is called an isosceles $n$-point if any three of its points are located in vertices of an isosceles triangle. Find all natural numbers for which there exist isosceles $n$-points.
2018 Yasinsky Geometry Olympiad, 5
The inscribed circle of the triangle $ABC$ touches its sides $AB, BC, CA$, at points $K,N, M$ respectively. It is known that $\angle ANM = \angle CKM$. Prove that the triangle $ABC$ is isosceles.
(Vyacheslav Yasinsky)
2021 Latvia TST, 2.1
Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$.
Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.
2014 Contests, 2 seniors
On the sides of triangle $ABC$, isosceles right-angled triangles $AUB, CVB$, and $AWC$ are placed. These three triangles have their right angles at vertices $U, V$ , and $W$, respectively. Triangle $AUB$ lies completely inside triangle $ABC$ and triangles $CVB$ and $AWC$ lie completely outside $ABC$. See the figure. Prove that quadrilateral $UVCW$ is a parallelogram.
[asy]
import markers;
unitsize(1.5 cm);
pair A, B, C, U, V, W;
A = (0,0);
B = (2,0);
C = (1.7,2.5);
U = (B + rotate(90,A)*(B))/2;
V = (B + rotate(90,C)*(B))/2;
W = (C + rotate(90,A)*(C))/2;
draw(A--B--C--cycle);
draw(A--W, StickIntervalMarker(1,1,size=2mm));
draw(C--W, StickIntervalMarker(1,1,size=2mm));
draw(B--V, StickIntervalMarker(1,2,size=2mm));
draw(C--V, StickIntervalMarker(1,2,size=2mm));
draw(A--U, StickIntervalMarker(1,3,size=2mm));
draw(B--U, StickIntervalMarker(1,3,size=2mm));
draw(rightanglemark(A,U,B,5));
draw(rightanglemark(B,V,C,5));
draw(rightanglemark(A,W,C,5));
dot("$A$", A, S);
dot("$B$", B, S);
dot("$C$", C, N);
dot("$U$", U, NE);
dot("$V$", V, NE);
dot("$W$", W, NW);
[/asy]