What is MidPoint Property of Triangle ?
Mid point property of triangle says : The line segment joining the midpoints of two sides of a triangle, is parallel to the third side
Let's understand this with the help of following diagram:
ABC is a triangle
X and Y are the mid points of sides AC and BC respectively
XY is the line segment which joins the midpoints of AC and BC sides
Now, Mid point property of triangle says that the line segment joining the midpoints of two sides of a triangle, is parallel to the third side. So in context of the above diagram, we get:
XY // AB
How to prove the midpoint property of triangle:
Before you understand how to prove this property, you are advised to read:
What is Transversal ?
What are Alternative Interior Angles of Transversal ?
What are Vertically Opposite Angles ?
What is Congruent Triangle ?
What are the Rules of Congruency ?
Observe the below diagram:
ABC is a triangle
X and Y are the mid points of sides AC and BC respectively, so we get:
AX = XC and CY = YB ..... (statement 1)
Now, from vertex B draw a line BL parallel to AC (as shown below)
Extend line XY such that it meets BL at Z (as shown below):
Now, BL // AC, so we can also say that:
AX // BZ ..... (statement 2)
Take two Triangles XYC and YZB (as highlighted in below diagram):
In △ CXY and △ YZB
∠ 1 = ∠ 2 (vertically opposite angles)
CY = YB (proved in statement 1)
∠ 3 = ∠ 4 (Alternative interior angles  AC // BL, CB is a transversal)
Therefore, by ASA rule of Congruency, we get:
△ CXY ≅ △ YZB
Since, △ CXY ≅ △ YZB, so we get:
XC = BZ (because corresponding sides of congruent triangles are equal) ..... (Statement 3)
From statement 1:
AX = XC
From statement 3:
XC = BZ
So, from statement 1 and 3, we get:
AX = BZ ..... (statement 4)
Now, observe Quadrilateral ABZX (as highlighted in below diagram):
From statement 2 and 4, we get:
AX // BZ
AX = BZ
Since AX and BZ are opposite side, so we get:
Quadrilateral ABZX is a parallelogram (because if opposite side of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram)
Now, in parallelogram ABZX, we get:
XZ // AB (because in parallelogram opposite side are parallel)
Or we can also write it as:
XY // AB
Hence proved, Mid point property of triangle : The line segment joining the midpoints of two sides of a triangle, is parallel to the third side
Now, since Mid point property is proved, so we get converse of this property i.e.:
The line drawn through the midpoint of one side of a triangle, parallel to another side bisects the third side.
