• Feb 2nd 2009, 09:59 AM
bigb
a) Let ABCD be a quadrilateral. Consider the points http://qaboard.cramster.com/Answer-B...4195000932.gifhttp://qaboard.cramster.com/Answer-B...4762508638.gif and prove that EFGH is a parallelogram.

b) Let (a,A), (b,B), (c,C), (d,D) be mass-points. Consider http://qaboard.cramster.com/Answer-B...6950005195.gif

Under which conditions on a,b,c,d are the points E,F,G,H a parallelogram?
• Feb 2nd 2009, 12:26 PM
Opalg
Hint: A parallelogram is a quadrilateral whose diagonals bisect each other (so the midpoint of one diagonal is equal to the midpoint of the other diagonal).
• Feb 2nd 2009, 02:54 PM
bigb
Quote:

Originally Posted by Opalg
Hint: A parallelogram is a quadrilateral whose diagonals bisect each other (so the midpoint of one diagonal is equal to the midpoint of the other diagonal).

I really still dont know how to do this problem. I drew an arbitrary quadrilateral and used the formula A+C=B+D for a parallelogram but still cant prove it.
• Feb 3rd 2009, 10:23 AM
Soroban
Hello, bigb!

Quote:

a) Let $\displaystyle ABCD$ be a quadrilateral.

Consider the points: .$\displaystyle E \:=\:\tfrac{1}{3}(A + 2B),\;F \:=\:\tfrac{1}{3}(2B + C),\;G \:=\:\tfrac{1}{3}(C + 2D),\;H \:=\:\tfrac{1}{3}(2D+A)$

Prove that EFGH is a parallelogram.

I must assume that those equations refer to the lengths of line segments,
and that the sides are divided in the ratio 1:2 or 2:1.

The diagram looks somthing like this:
Code:

            A          E          B             o - - - - - o - - - - - o           *        *      *        *           *    *              *      *         *  *                      *    *       H o                              *  *       *    *                              * *       *        *                              o F     *              *                      *    *     *                  *              *          *   *                        *      *                *   o - - - - - - - - - - - - - - o - - - - - - - - - - - o   D                            G                      C
Draw diagonal $\displaystyle BD.$

Consider $\displaystyle \Delta AEH\text{ and }\Delta ABD$
. . $\displaystyle \angle EAH \,=\,\angle BAD$
. . $\displaystyle AE:AB \,=\,1:3 \quad\Rightarrow\quad AE \,=\,\tfrac{1}{3}AB$
. . $\displaystyle AH:AD \,=\,1:3 \quad\Rightarrow\quad AH \,=\,\tfrac{1}{3}AD$
Hence: .$\displaystyle \Delta AEH \sim \Delta ABD$
. . $\displaystyle HE:DB \,=\,1:3 \quad\Rightarrow\quad HE \,=\,\tfrac{1}{3}DB$ .[1]
. . $\displaystyle \angle AHE\,=\,\angle ADB \quad\Rightarrow\quad HE \parallel DB$ .[2]

Consider $\displaystyle \Delta CFG\text{ and }\Delta CBD$
. . $\displaystyle \angle FCG \,=\,\angle BCD$
. . $\displaystyle CF:CB \,=\,1:3 \quad\Rightarrow\quad CF \,=\,\tfrac{1}{3}CB$
. . $\displaystyle CG:CD \,=\,1:3 \quad\Rightarrow\quad CG \,=\,\tfrac{1}{3}CD$
Hence: .$\displaystyle \Delta CFG \sim \Delta CBD$
. . $\displaystyle FG:DB \,=\,1:3\quad\Rightarrow\quad FG \,=\,\tfrac{1}{3}DB$ .[3]
. . $\displaystyle \angle FGC \,=\,\angle BDC \quad\Rightarrow\quad FG \parallel DB$ .[4]

From [1] and [3]: .$\displaystyle HE\,=\,FG$

From [2] and [4]: .$\displaystyle HE \parallel FG$

Theorem: If two sides of a quadrilateral are equal and parallel,
. . . . . . . the quadrilateral is a parallelogram.

Therefore, $\displaystyle EFGH$ is a parallelogram.