Orthocentric System From Rectangle

Orthocentric System From Rectangle: What Is This About?
A Mathematical Droodle

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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.

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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

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

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