In ` A B C ,D` is the mid-point of `A B ,P` is any point of `B CdotC Q P D` meets `A B` in `Q` . Show that `a r( B P Q)=1/2a r( A B C)dot` TO PROVE : `a r( B P Q)=1/2a r( A B C)` CONSTRUCTION : Join CD.

Explanation

`"CD is the median to side AB"`

`"ar"(/_\BCD)="ar"(/_\DAC)->"(1)"`

`"[A median divides a "/_\" into two " /_\ "s of equal area]"`

`/_\ PDC and /_\PDQ, " they lie on same base PD and are between same parallels PD and CQ:"`

`therefore "ar"(/_\PDC)="ar"(/_\PDQ)->"(2)"`

`"ar(BCD) + ar(DAC) = ar"(/_\ABC)`

`therefore "2 ar(BCD) = ar"(/_\ABC)-> "[from eq (1)]"`

`"ar"(/_\BCD) = 1/2 ar (/_\ABC)->"eq(3)"`

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