यदि एक त्रिभुज ABC की भुजाओं BC, CA और AB के मध्य बिन्दु क्रमशः D, E और F हैं, तो निर्देशांक ज्यामिति से सिद्ध कीजिए कि
`Delta DEF` का क्षेत्रफल `=1/4 ` (`Delta ABC` का क्षेत्रफल )

Explanation

माना `A(x_(1),y_(1)),B(x_(2),y_(2)),C(x_(3),y_(3))`त्रिभुज ABC के शीर्ष हैं, तो D, E और F के निर्देशांक

क्रमशः `((x_(2)+x_(3))/(2),(y_(2)+y_(3))/(2))((x_(1)+x_(3))/(2),(y_(1)+y_(3))/(2))`और `((x_(1)+x_(2))/(2), (y_(1)+y_(2))/(2))`हैं ।
अब
`Delta_(1)=Delta ABC` का क्षेत्रफल `=1/2 |x_(1)(y_(2)-y_(3))+x_(2)(y_(3)-y_(1))+x_(3)(y_(1)-y_(2))|`
`Delta_(2)=Delta DEF` का क्षेत्रफल `=1/2 |((x_(2)+x_(3))/(2))((y_(1)+y_(3))/(2)-(y_(1)+y_(2))/(2))+((x_(1)+x_(3))/(2))((y_(1)+y_(2))/(2)-(y_(2)+y_(3))/(2))+((x_(1)+x_(2))/(2))((y_(2)+y_(3))/(2)-(y_(1)+y_(3))/(2))|`
`rArrDelta_(2)=1/8|(x_(2)+x_(3))(y_(3)-y_(2))+(x_(1)+x_(3))(y_(1)-y_(3))+(x_(1)+x_(2))(y_(2)-y_(1))|`
`rArr Delta_(2)=1/8 |x_(1)(y_(1)-y_(3)+y_(2)-y_(1))+x_(2)(y_(3)-y_(2)+y_(2)-y_(1))+x_(3)(y_(3)-y_(2)+y_(1)-y_(3))|`
`rArr Delta_(2)=1/8 |x_(1)(y_(2)-y_(3))+x_(2)(y_(3)-y_(1))+x_(3)(y_(1)-y_(2))|`
`rArr Delta_(2)=1/4 (DeltaABC "का क्षेत्रफल ")=1/4 Delta_(1)`
अतः `Delta DEF` का क्षेत्रफल `=1/4 `( `Delta ABC` का क्षेत्रफल )