Problem
GEO-B3-M04-P005 The Simson Line
B. New Original Problem. A point \(P\) lies on the circumcircle of triangle \(ABC\). Let \(A_1,B_1,C_1\) be the projections of \(P\) onto the lines \(BC,CA,AB\). Prove that \(A_1,B_1,C_1\) are collinear.
C. Hint 1. Use the circles with diameters \(PA,PB,PC\).
D. Hint 2. Compare directed angles at \(B_1\).
E. Full Solution.
The right angles give cyclic quadrilaterals \(P,A_1,C,B_1\) and \(P,B_1,A,C_1\). Hence \(\angle A_1B_1C=\angle A_1PC\), and \(\angle C_1B_1A=\angle C_1PA\) in the directed sense.
Since \(A,B,C,P\) lie on one circle, angles subtending the corresponding arcs give equality of these two directed angles. Thus the rays \(B_1A_1\) and \(B_1C_1\) lie on one line. Therefore \(A_1,B_1,C_1\) are collinear.
The main theorem of the module; directed angles are important.