Mastering Geometry with Apollonius Theorem and Cosine Rule: A Problem from the RMO 2005

In this insightful video, we explore various geometric concepts through a problem from the RMO 2005 (Regional Math Olympiad). The focus of the problem is a convex quadrilateral, and through this, several important geometric theorems and techniques are applied.

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Here's what we cover:

• Convex Quadrilateral Definition: The video begins by clarifying the concept of a convex quadrilateral, comparing it with real-world examples like the map of India. A quadrilateral is convex if the shortest path between any two points inside the shape remains within the quadrilateral.
• Midpoint Theorem Application: The midpoint theorem is used to demonstrate that two equilateral triangles, formed by the midpoints of the quadrilateral, are congruent. This is done by proving that the line joining the midpoints of two sides is half the length of the third side.
• Introduction to Apollonius’ Theorem: Apollonius’ theorem is introduced as a powerful tool for calculating medians in a triangle in terms of its sides. The theorem is applied twice, in two different triangles, to show that two sides of the quadrilateral are equal, leading to the conclusion that it is a rhombus.
• Isosceles Triangle Property: By using the fact that if two medians of a triangle are equal, the triangle is isosceles, the video illustrates how this can be applied to show that the quadrilateral is a rhombus.
• Cosine Rule and Geometry Bashing: The video demonstrates how to use the cosine rule in conjunction with algebraic methods (referred to as "bashing") to solve for unknown sides and angles of the rhombus. The cosine rule is applied to triangles within the quadrilateral to find relationships between the angles and sides.
• Centroid Properties: Centroid properties are used to show that medians divide the triangle in a 2:1 ratio, which is crucial for the final calculations involving the diagonals of the rhombus.
• Final Challenge: After determining that the quadrilateral is a rhombus, viewers are left with a challenge: calculate the angles of the rhombus, given that one of the diagonals is equal to one of its sides.

This problem elegantly ties together theorems like Apollonius' theorem, the midpoint theorem, and the cosine rule, providing students with multiple tools to tackle complex geometry problems.