This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 3

2011 German National Olympiad, 3
Tags: geometry , altitude , circumcircle , orthic triangle
Let $ABC$ be an acute triangle and $D$ the foot of the altitude from $A$ onto $BC$. A semicircle with diameter $BC$ intersects segments $AB,AC$ and $AD$ in the points $F,E$ resp. $X$. The circumcircles of the triangles $DEX$ and $DXF$ intersect $BC$ in $L$ resp. $N$ other than $D$. Prove $BN=LC$.

1988 Bundeswettbewerb Mathematik, 3
Tags: geometry , perimeter , orthic triangle
Prove that all acute-angled triangles with the equal altitudes $h_c$ and the equal angles $\gamma$ have orthic triangles with same perimeters.

1909 Eotvos Mathematical Competition, 3
Tags: geometry , orthic triangle
Let $A_1, B_1, C_1$, be the feet of the altitudes of $\vartriangle ABC$ drawn from the vertices $A, B, C $ respectively, and let $M$ be the orthocenter (point of intersection of altitudes) of $\vartriangle ABC$. Assume that the orthic triangle (i.e. the triangle whose vertices are the feet of the altitudes of the original triangle) $A_1$,$B_1$,$C_1$ exists. Prove that each of the points $M$, $A$, $B$, and $C$ is the center of a circle tangent to all three sides (extended if necessary) of $\vartriangle A_1B_1C_1$. What is the difference in the behavior of acute and obtuse triangles $ABC$?