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Explanation

The applet below provides an illustration to a problem from an outstanding collection by T. Andreescu and R. Gelca:

 Let ABCD be a rectangle and let P be a point on its circumcircle, different from any vertex. Let X, Y, Z, and W be the projections of P onto the lines AB, BC, CD, and DA, respectively. Prove that the points X, Y, Z, W form an orthocentric system.

### This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

 What if applet does not run?

Since obviously, XZ is orthogonal to YW, suffice it to show that XY⊥ ZW.

In rectangle BXPY, angles BPX and BYX are equal and so are angles DWZ and DPZ in rectangle DZPW. Since P lies on the a circle with diameter BD, ∠BPD = 90° and therefore

 ∠BPX + ∠DWZ = 90°.

This means that lines XY and ZW form equal angles with two perpendicular lines (e.g., with BY and WY) and are thus perpendicular to each other.

### References

1. T. Andreescu, R. Gelca, Mathematical Olympiad Challenges, Birkhäuser, 2004, 5th printing, 1.3.3 (p. 8)