Thread: diagonals of a rectangle and a square

1. 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?

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