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This document explores the relationships between midpoints of a triangle and their consequences using segments defined by midpoints. Given points A, B, and C, we denote midpoints X, Y, and Z. We establish equalities like AX = BX, AZ = CZ, and BY = CY, which reveal how these midpoints divide the sides of triangle ABC. The document incorporates properties of congruence and similarity, demonstrating relationships through the Medial Triangle Theorem and various equality properties. This study highlights the critical concepts in triangle geometry and their proofs.
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x is the midpoint of AB z is the midpoint of AC y is the midpoint of BC AX =BX AZ = CZ BY = CY AB = 2ZY AC = 2XY BC = 2XZ 1/2 AB = ZY 1/2 AC = XY 1/2 BC = XZ 1/2 AB = AX = XB 1/2 AC = AZ = CZ 1/2 BC = BY = CY ZY = AX = XB XY = AX = XB XY = AZ = CZ XZ = CY = BY AXZ = YXZ = XBY = ZYC XYZ similar ABC 4 XYZ = ABC XYZ = ABC/4 Given Definition of a midpoint Medial Triangle Theorem Division Property of Equality Definition of a midpoint Substitution Side Side Side Congruence Side Side Side Similarity Substitution Division Property of Equality Statement Reason
