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48th Austrian Mathematical Olympiad

Austria geometry

Problem

In the isosceles triangle with we denote by the foot of the altitude through . The midpoint of is denoted by . The line intersects in . Prove that the length of is three times that of .

problem
Figure 4: Problem 16
Solution
We consider the centroid of triangle , which lies on the axis ; see Figure 4. The centroidal axis bisects the segment , thus is parallel to by the intercept theorem. Since the centroid divides the centroidal axis in the ratio , we have the same division ratio on the parallel line , i.e. divides in the ratio .

Techniques

Triangle centers: centroid, incenter, circumcenter, orthocenter, Euler line, nine-point circle