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.

  • 3
    $\begingroup$ Hint: $\;\frac{HA}{2} \cdot HH_a\,$ is the power of point $H$ with respect to the 9-point circle. $\endgroup$
    – dxiv
    Jun 23 at 3:07
  • $\begingroup$ @dxiv ohhh... then $\frac{HA}{2} \cdot HH_A = \frac{HB}{2} \cdot HH_B = \frac{HC}{2} \cdot HH_C$. Thanks! $\endgroup$
    – Luigui1729
    Jun 23 at 3:57
  • 1
    $\begingroup$ Right. Or, for an alternate proof, chase pairs of similar right triangles. $\endgroup$
    – dxiv
    Jun 23 at 4:09


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