Problem
GEO-B2-M09-P008 Coordinates of the Orthocenter
#8
★★☆☆☆ Level 2 of 5
In triangle \(A(0,0)\), \(B(p,0)\), \(C(q,r)\), where \(p,r\ne 0\), find the coordinates of the orthocenter.
The altitude from \(C\) is vertical. Intersect it with the altitude from \(B\).
Since \(AB\) is horizontal, the altitude from \(C\) has equation \(x=q\). Line \(AC\) has slope \(\frac rq\) when \(q\ne 0\), so the altitude from \(B\) has slope \(-\frac qr\) and equation \(y=-\frac qr(x-p)\). At \(x=q\), we get \(y=\frac{q(p-q)}{r}\). If \(q=0\), the same formula gives \(y=0\). Hence \(H\left(q,\frac{q(p-q)}{r}\right)\).
A useful formula for later orthocenter problems.