Pythagorean Theorem in an Orthodiagonal Quadrilateral

Let there be two squares: APBM_c and C₁M_cC₂Q with a common vertex M_c. Rotation through 90^o in the clockwise direction around M_c, moves C₂M_c into C₁M_c and AM_c into BM_c. This implies that DAM_cC₂ rotates into DBM_cC₁ so that AC₂ and BC₁ are orthogonal. Quadrilateral ABC₂C₁ is thus orthodiagonal (i.e., having orthogonal diagonals) and the configurations is remarkable: the red and blue squares add up to the same area. The important point to note is that the sum of the areas of the original squares APBM_c and C₁M_cC₂Q is half this quantity.

In a special case where M_c coincides with the point of intersection of the diagonals we get a proof of the Pythagorean theorem:

Because of the resulting symmetry, the red squares are equal. Therefore, the areas of APBM_c and C₁M_cC₂Q add up to that of a red square!

---

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 http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

---

Orthodiagonal Quadrilaterals

1. Invariance in Orthodiagonal Quadrilaterals
2. Orthodiagonal and Cyclic Quadrilaterals
3. Classification of Quadrilaterals
4. Pythagorean Theorem in an Orthodiagonal Quadrilateral
5. Easy Construction of Bicentric Quadrilateral
6. Easy Construction of Bicentric Quadrilateral II

|Up| |Contact| |Front page| |Contents| |Geometry| |Store|