Competition problem — Geometry — Olympiad Maths Prep

Geometry Difficulty 4.9 AIME Prove it Austria

In the isosceles triangle $ABC$ with $\overline{AC} = \overline{BC}$ we denote by $D$ the foot of the altitude through $C$ . The midpoint of $CD$ is denoted by $M$ . The line $BM$ intersects $AC$ in $E$ . Prove that the length of $AC$ is three times that of $CE$ .

Figure 4: Problem 16

Solution

We consider the centroid $S$ of triangle $DBC$ , which lies on the axis $BM$ ; see Figure 4. The centroidal axis $DS$ bisects the segment $BC$ , thus $DS$ is parallel to $AC$ by the intercept theorem. Since the centroid divides the centroidal axis in the ratio $2 : 1$ , we have the same division ratio on the parallel line $AC$ , i.e. $E$ divides $AC$ in the ratio $2 : 1$ .

Looking for a route rather than an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.