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Math Help - diagonals of a rectangle and a square

  1. #1
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    diagonals of a rectangle and a square

    If we have a square ABCD and a rectangle AEFD. Let P be the intersection of AC and ED, Q the intersection of AF and BD. How can we prove that the line PQ is parallel to AD?
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  2. #2
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    Hello, geo_math!

    We have a square ABCD and a rectangle AEFD.
    Let P be the intersection of AC and ED, Q the intersection of AF and BD.

    Prove that PQ is parallel to AD.
    Code:
        Q *
          : * *
          :   *   *
          :     *     *
          :       *       *
          :         *      θ  *   A
          :         B * - - - - - * - - - - - - - - - - - * E
          :           | *     α * |   *  θ         θ  *   |
          :           |   *   * α |       *       *       |
          :           |     *     |           *           |
          :           |   * R * α |       *       *       |
          :           | *     α * |   *  θ         θ  *   |
          :         C * - - - - - * - - - - - - - - - - - * F
          :         *      θ  *  D
          :       *       *
          :     *     *
          :   *   *
          : * *
        P *

    Let AC and BD intersect at R.
    Note that all angles labeled \theta are equal.
    . . And all angles labeled \alpha are 45.
    Further note that: . AC \perp BD\:\text{ and }\:RA = RD

    In right triangles PRD and QRA:\!\;\;\angle PDR \,=\,\angle QAR \,=\,\theta + 45^o,\;\;RD = RA

    Hence: . \Delta PRD \cong \Delta QRA \quad\Rightarrow\quad PR \,=\,QR

    Then \Delta PRQ is an isosceles right triangle: . \angle PQR \,=\,45^o

    Since \angle QDA = 45^o,\;\;PQ \parallel AD\quad\text{(alternate-interior angles)}

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