IIT JEE , AIEEE Forum - Mathematics , Analytical Geometry

gotiit

question: prove that ; if mid points of adjacent sides of any quadrilateral are joined with straight lines, then the inner quadrilateral so formed is a parallelogram .

srujana

$Let\;A(x_1,y_1),B(x_2,y_2),C(x_3,y_3),D(x_4,y_4),\;be\;the\;vertices\;of\;the\;quadrilateral.\\\\$

$midpoint\;of\;AB,\;E=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

$similarly,let\;the\;midpoints\;of\;BC,CD,DA\;be\;F,G,H\;respectively.$

$midpoint\;of\;EF\;=({\frac{x_1+x_2+x_3+x_4}{4},\frac{y_1+y_2+y_3+y_4}{4}})$

$similarly\ midpoint\;of\;GH\;=({\frac{x_1+x_4+x_3+x_2}{4},\frac{y_1+y_4+y_3+y_2}{4}})$

$EF\ and\ GH\ have\ a\ common\ midpoint\ hence\ diagonals\ bisect\ each\ other\ i.e,EFGH\ is\ a\ llelogram.$

oneyeartogo

I think u have made a typing mistake. EF is a side not diagonal. It should be EG. Otherwise excellent proof.

srujana

yep EG and FH...apologies for the typo :)

srujana

Alternate method:

We have the theorem, a line drawn through the midpoints of two sides of a triangle is always parallel to the third side(midpoint theorem).

consider the triangles ABC,ADC.

Let E,F,G,H be the midpoints of AB,BC,CD,DA respectively.

using midpoint theorem, EF||AC||GH

EG||BD||FG

Hence, E,F,G,H form a llelogram.

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