Consider a non-isosceles triangle, pick a vertex. Assume that the median and the altitude passing through this vertex are isogonal conjugates (i.e. symmetric w.r.t. the bisector of the angle). Prove that the triangle is right-angled in this vertex.

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Eleanor Luna · Accepted Answer

Let   △  A  B  C have the desired property at   A. Let   M,   P,   N be the points where the median, angle bisector, and altitude from   A meet side             B      C        ¯  .Since  ∠  B  A  P  ≅  ∠  C  A  P    and    ∠  M  A  P  ≅  ∠  N  A  Pwe have  ∠  B  A  M  ≅  ∠  C  A  N  =      C    ′      and    ∠  C  A  M  ≅  ∠  B  A  N  =      B    ′  where       B    ′   and       C    ′   are respective complements of   B and   C. Invoking the Law of Sines in   △  B  A  M and   △  C  A  M,                            |                              A            M                    ¯                      |                                      sin            ⁡                          C              ′                                            sin            ⁡            B                          =              |                              B            M                    ¯                      |            ⏟              △      B      A      M        =      1    2    |            B      C        ¯    |  =                            |                              C            M                    ¯                      |        =              |                              A            M                    ¯                      |                                      sin            ⁡                          B              ′                                            sin            ⁡            C                              ⏟              △      C      A      M      Therefore,                    sin        ⁡        B        sin        ⁡                  B          ′                =        sin        ⁡        C        sin        ⁡                  C          ′                                                    ⟹                        sin        ⁡        B        cos        ⁡        B        =        sin        ⁡        C        cos        ⁡        C                                                  ⟹                        sin        ⁡        2        B        =        sin        ⁡        2        C                                                  ⟹                        B        =        C                or                2        B        =        π        −        2        C                                                  ⟹                        B        =        C                or                B        +        C        =        π              /        2                                                  ⟹                        B        =        C                or                A        =        π              /        2

Consider a non-isosceles triangle, pick a vertex. Assume that the

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