Let ABC be an acute angled triangle with circumcenter O.

Kallie Arroyo 2022-06-02 Answered
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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Answers (1)

a1p2ung1ls6t7
Answered 2022-06-03 Author has 6 answers
Step 1
First 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'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

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