Timeless Theorems of Mathematics/Mid Point Theorem

thumb|In this right triangle, and according to the Mid Point Theorem The midpoint theorem is a fundamental concept in geometry that establishes a relationship between the midpoints of a triangle's sides. This theorem states that when you connect the midpoints of two sides of a triangle, the resulting line segment is parallel to the third side. Additionally, this line segment is precisely half the length of the third side. Proof Statement In a triangle, if a line segment connects the midpoints of two sides, then this line segment is parallel to the third side and half its length. Proof with the help of Congruent Triangles thumb|The construction for the mid-point theorem's proof with similar triangles Proposition: Let and be the midpoints of and in the triangle . It is to be proved that, and; . Construction: Add and , extend to as , and add and . Proof: [1] In the triangles and ; [Given] ; [According to the construction] ; [Vertical Angles] ∴ ; [Side-Angle-Side theorem] So, ∴ Or, and Therefore, is a parallelogram. ∴ or [2] Or Or, [As, ] Or, Or, ∴ In the triangle and , where and are the midpoints of and . [Proved] Proof with the help of Coordinate Geometry Proposition: Let and be the midpoints of and in the triangle , where the coordinates of are . It is to be proved that, and Proof: [1] The distance of the segment The midpoint of and is . In the same way, The midpoint of and is ∴ The distance of ; [As, ] [2] The slope of The slope of ; [As, ] Therefore, ∴ In the triangle and , where and are the midpoints of and . [Proved]