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Pedal triangle

Found it while further playing with this question.

Let $P$ be a point inside an acute triangle $ABC$ (whether it works for obtuse ones - I doubt it). Let $A'B'C'$ be the pedal triangle of $P$. Now assume that (with given constants $\lambda,\mu$) $\psi=\lambda*\alpha+\mu*\pi$, and analogous with $\beta,\gamma$ and the other angles at $P$. ($\lambda+3*\mu=2$, obviously.)

Then also $\phi=\nu*\alpha+\mu*\pi$, and again cyclically. ($\nu+3*\mu=1$, obviously.)

Random (well-known) examples.

  • Let $P$ be the incenter. $\lambda=1/2,\mu=1/2,\nu=-1/2$.
  • Let $P$ be the orthocenter. $\lambda=-1,\mu=1,\nu=-2$.
  • Let $P$ be the circumcenter. $\lambda=2,\mu=0,\nu=1$.

I probably could solve this myself in a math exam, but my head spins from all the angles and I have a hard chess match today :-) Also, I've got a hunch that this can be generalized vastly.

Hauke Reddmann
  • 735
  • Instead of a reference to an other question, (which is also hard to digest regarding notations,) please introduce all the objects one by one. Please make clear what is given - in the order things are given - and define all variables occurring in the text. What are the letters $\alpha,\beta,\gamma$? What are $\lambda,\mu$? (One cannot give $P,\alpha,\beta,\gamma,\lambda,\mu$ in the same time. If $P$ is given, how are $\lambda,\mu$ introduced? Note that $P$ is quite general above.) And which is explcitly the question? As a matter of typing, that $*$ is rather disturbing. Do we really need it? – dan_fulea Mar 24 '22 at 19:07

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