Problem
GEO-B1-M07-P016 A Circle on Altitudes
#16
★★★☆☆ Level 3 of 5
In triangle \(ABC\), points \(D\) and \(E\) are the feet of the altitudes from \(B\) and \(C\). Prove that \(B,C,D,E\) lie on one circle.
Draw the circle with diameter \(BC\).
Since \(BD\perp AC\), and \(D\in AC\), we get \(\angle BDC=90^\circ\). Since \(CE\perp AB\), and \(E\in AB\), we get \(\angle BEC=90^\circ\). Hence points \(D\) and \(E\) lie on the circle with diameter \(BC\). Therefore \(B,C,D,E\) lie on one circle.
Connects the circle construction with altitudes.