# Is it true that $AH*HH_A=BH*HH_B=CH*HH_C$ on $\triangle ABC$ with orthocenter $H$ and bases intersections $H_A, H_B, H_C$?

I was goofing around in geogebra and it appears that if we have $$\triangle ABC$$ with orthocenter $$H$$ and where the bases of the heights are $$H_A, H_B, H_C$$, then the following holds true: $$AH*HH_A=BH*HH_B=CH*HH_C$$ I couldn't find anything in the internet... is this actually true or am I mistaken? If true, what would be a nice proof? Thanks.

• Hint: $\;\frac{HA}{2} \cdot HH_a\,$ is the power of point $H$ with respect to the 9-point circle.
– dxiv
Jun 23 at 3:07
• @dxiv ohhh... then $\frac{HA}{2} \cdot HH_A = \frac{HB}{2} \cdot HH_B = \frac{HC}{2} \cdot HH_C$. Thanks! Jun 23 at 3:57
• Right. Or, for an alternate proof, chase pairs of similar right triangles.
– dxiv
Jun 23 at 4:09