Let ABC be an acute angled triangle with circumcenter O. A circle passing through A and O intersects AB, AC at P, Q respectively. Show that the orthocentre of triangle OPQ lies on the side BC.

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a1p2ung1ls6t7 · Accepted Answer

Step 1First of all, let us call   α  =  ∠  B  A  C and   β  ,  γ the other two. Using that APOQ is a cyclic quadrilateral, we have that   ∠  Q  P  O  =  ∠  Q  A  O  =      π    2    −  β , and similarly   ∠  P  Q  O  =      π    2    −  γ .Now, take the circunference through O,P,B, and let R be the intersection with the side             B      C        ¯   . It is not so difficult to check that CQOR is a cyclic quadrilateral (and is an excellent excercise if you haven&#039;t done it before). This allows us to compute the angles  ∠  R  P  O  =  ∠  O  Q  R  =      π    2    −  α  . .In particular, this shows that             P      O        ¯    ⊥            R      Q        ¯   and             Q      O        ¯    ⊥            P      R        ¯   , showing that R is the orthocenter of PQO

Let ABC be an acute angled triangle with circumcenter O.

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