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D,E,F are the midpoints of the sides bar...

D,E,F are the midpoints of the sides `bar(BC),bar(CA)` and `bar(AB)` respectively of the triangle ABC. If P is any point in the plane of the traingle , show that `bar(PA)+bar(PB)+bar(PC)= bar(PD)+bar(PE)+bar(PF)`.

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D, E, F are the midpoints of the sides bar(BC), bar(CA) " and " bar(AB) respectively of the triangle ABC. If P is any point in the plane of the triangle, show that vec(PA)+vec(PB)+vec(PC)=vec(PD)+vec(PE)+vec(PF) .

Let D, E, F are the midpoints of the sides bar(BC), bar(CA) " and " bar(AB) of the triangle ABC. Prove that vec(AD)+vec(BE)+vec(CF)=vec(0) .

The coordinatea of A,B and C of the DeltaABC are (3,1), (9,7) and (-3,7) respectively. If D,E and F are the mid-point of the sides bar(BC),bar(CA)andbar(AB) respectively, then find the area of the DeltaDEF. Also show that DeltaABC=4DeltaDEF.

ABCD is a parallelogram, P and Q are the mid-points of the sides bar(AB) " and " bar(DC) respectively. Show that bar(DP) " and " bar(BQ) trisect bar(AC) and are trisected by bar(AC) .

Prove the following boolean relations. bar(AB + BC + CA) = bar(A)bar(B) + bar(B)bar(C ) +bar(C)bar(A)

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Prove the following boolean relations. bar(bar(A)+AB+bar(AB)) = 0

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In the adjacent figure DeltaABC is isosceles as bar(AB) = bar(AC), bar(BA) and bar(CA) are produced to Q and P such that bar(AQ) = bar(AP) . . Show that bar(PB) = bar(QC) .

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