Problem
GEO-B2-M09-P020 Two Altitudes and One Circle
#20
★★★★☆ Level 4 of 5
In triangle \(A(0,0)\), \(B(1,0)\), \(C(u,v)\), \(v\ne 0\), point \(D\) is the foot of the altitude from \(C\) to \(AB\), and \(E\) is the foot of the altitude from \(B\) to \(AC\). Prove that \(B,C,D,E\) lie on one circle.
It is enough to prove that \(\angle BDC=\angle BEC=90^\circ\).
Since \(D\) is the foot of the altitude from \(C\), \(CD\perp DB\), so \(\angle BDC=90^\circ\). Since \(E\) is the foot of the altitude from \(B\), \(BE\perp CE\), so \(\angle BEC=90^\circ\). Therefore points \(D\) and \(E\) lie on the circle with diameter \(BC\). Hence \(B,C,D,E\) are concyclic.
One may additionally ask the student to find the coordinates of \(E\) by projection onto \(AC\).