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Thread: 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 $\displaystyle ABCD$ and a rectangle $\displaystyle AEFD$.
    Let $\displaystyle P$ be the intersection of $\displaystyle AC$ and $\displaystyle ED$, $\displaystyle Q$ the intersection of $\displaystyle AF$ and $\displaystyle BD.$

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

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

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

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

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

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

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