Menu Close

Consider-the-quadrilateral-ABCD-The-points-M-N-P-and-Q-are-the-midpoints-of-the-sides-AB-BC-CD-and-DA-Let-X-AP-BQ-Y-BQ-CM-Q-CM-DN-and-T-DN-AP-Prove-that-XYZT-AQX-BMY-




Question Number 16946 by Tinkutara last updated on 28/Jun/17
Consider the quadrilateral ABCD.  The points M, N, P and Q are the  midpoints of the sides AB, BC, CD  and DA.  Let X = AP ∩ BQ, Y = BQ ∩ CM,  Q = CM ∩ DN and T= DN ∩ AP.  Prove that [XYZT] = [AQX] + [BMY]  + [CNZ] + [DPT].
$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{quadrilateral}\:{ABCD}. \\ $$$$\mathrm{The}\:\mathrm{points}\:{M},\:{N},\:{P}\:\mathrm{and}\:{Q}\:\mathrm{are}\:\mathrm{the} \\ $$$$\mathrm{midpoints}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sides}\:{AB},\:{BC},\:{CD} \\ $$$$\mathrm{and}\:{DA}. \\ $$$$\mathrm{Let}\:{X}\:=\:{AP}\:\cap\:{BQ},\:{Y}\:=\:{BQ}\:\cap\:{CM}, \\ $$$${Q}\:=\:{CM}\:\cap\:{DN}\:\mathrm{and}\:{T}=\:{DN}\:\cap\:{AP}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\left[{XYZT}\right]\:=\:\left[{AQX}\right]\:+\:\left[{BMY}\right] \\ $$$$+\:\left[{CNZ}\right]\:+\:\left[{DPT}\right]. \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *