Problem
GEO-B1-M05-P023 Two Circles in Altitudes
#23
★★★★☆ Level 4 of 5
In acute triangle \(ABC\), altitudes \(BD\) and \(CE\) meet at point \(H\), where \(D\in AC\), \(E\in AB\). Prove that points \(A,D,H,E\) lie on one circle, and points \(B,C,D,E\) also lie on one circle.
Find two pairs of right angles.
Since \(BD\perp AC\), we have \(\angle ADH=90^\circ\). Since \(CE\perp AB\), we have \(\angle AEH=90^\circ\). Thus points \(D\) and \(E\) see segment \(AH\) under a right angle, so \(A,D,H,E\) lie on the circle with diameter \(AH\). Also, \(\angle BDC=90^\circ\) and \(\angle BEC=90^\circ\), so \(D\) and \(E\) see segment \(BC\) under a right angle. Therefore \(B,C,D,E\) lie on the circle with diameter \(BC\).
A strong training example: one configuration gives two different circles.