If D, E and F are mid-points of the sides BC, CA and AB respectively of a △ABC, prove that AD and FE bisect each other.

△ABC with D, E and F as mid-points of the sides BC, CA and AB is shown below: D and E are midpoints of BC and CA respectively, DE = AB and DE || AB or DE || AF .......(1) Since, F is midpoint of AB, AF = AB ∴ AF = DE .......(2) F and D are midpoints of AB and BC respectively, FD = AC and FD || AC or FD || AE .......(3) Since, E is midpoint of AC, AE = AC ∴ FD = AE .......(4) From 1, 2, 3 and 4 we get, DE || AF, AF = DE and FD || AE, FD = AE. Hence, AEDF is a parallelogram. ∴ AD and EF bisect each other. Hence, AD and EF bisect each other.

In the adjoining figure, ABCD is a parallelogram and E is mid-point of AD. DL || EB meets AB produced at F. Prove that B is mid-point of AF and EB = LF.

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In △ABC, D and E are mid-points of the sides AB and AC respectively. Through E, a straight line is drawn parallel to AB to meet BC at F. Prove that BDEF is a parallelogram. If AB = 8 cm and BC = 9 cm, find the perimeter of the parallelogram BDEF.

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ABCD is a rhombus with P, Q and R as mid-points of AB, BC and CD respectively. Prove that PQ ⊥ QR.

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The diagonals of a quadrilateral ABCD are perpendicular. Show that the quadrilateral formed by joining the mid-points of its adjacent sides is a rectangle.

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Mathematics

If D, E and F are mid-points of the sides BC, CA and AB respectively of a △ABC, prove that AD and FE bisect each other.

Mid-point Theorem

Related Questions

In the adjoining figure, ABCD is a parallelogram and E is mid-point of AD. DL || EB meets AB produced at F. Prove that B is mid-point of AF and EB = LF.

In △ABC, D and E are mid-points of the sides AB and AC respectively. Through E, a straight line is drawn parallel to AB to meet BC at F. Prove that BDEF is a parallelogram. If AB = 8 cm and BC = 9 cm, find the perimeter of the parallelogram BDEF.

ABCD is a rhombus with P, Q and R as mid-points of AB, BC and CD respectively. Prove that PQ ⊥ QR.

The diagonals of a quadrilateral ABCD are perpendicular. Show that the quadrilateral formed by joining the mid-points of its adjacent sides is a rectangle.