geometry problems from Austrian Junior Regional Competition with aops links in the names
collected inside aops here
2019 - 2021
A square $ABCD$ is given. Over the side $BC$ draw an equilateral triangle $BCD$ on the outside. The midpoint of the segment $AS$ is $N$ and the midpoint of the side $CD$ is $H$. Prove that $\angle NHC = 60^o$.
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(Karl Czakler)
Given is an isosceles trapezoid $ABCD$ with $AB \parallel CD$ and $AB> CD$. The The projection from $D$ on $ AB$ is $E$. The midpoint of the diagonal $BD$ is $M$. Prove that $EM$ is parallel to $AC$.
(Karl Czakler)
A triangle $ABC$ with circumcenter $U$ is given, so that $\angle CBA = 60^o$ and $\angle CBU = 45^o$ apply. The straight lines $BU$ and $AC$ intersect at point $D$. Prove that $AD = DU$.
(Karl Czakler)
A semicircle is erected over the segment $AB$ with center $M$. Let $P$ be one point different from $A$ and $B$ on the semicircle and $Q$ the midpoint of the arc of the circle $AP$. The point of intersection of the straight line $BP$ with the parallel to $P Q$ through $M$ is $S$. Show that $PM = PS$ holds.
(Karl Czakler)
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