# Integer triangle altitudes

Geometry Level 3

In triangle $$ABC,$$ the feet of the perpendiculars from $$A, B, C$$ are $$D, E, F,$$ respectively, and the following holds: $AH \le BH \le CH,$ where$$H$$ is the orthocenter of the triangle.

Find the specific triangle in which the segments $$AH, BH, CH, HD, HE, HF$$ all have integer lengths, $$AH$$ and $$HD$$ have an integer geometric mean, and $$AD+BE+CF$$ is as small as possible.

Then, enter the concatenated value of $$AH, HD, BH, HE, CH, HF$$.

×