Erdos-Mordell Inequality

This applet demonstrates the Erdos-Mordell Inequality, a beautiful classical result in triangle geometry. The inequality states that for any point located inside a triangle, the sum of its distances to the three vertices is greater than or equal to twice the sum of its distances to the three sides. In notation, if P is a point inside triangle ABC with distances PA, PB, and PC to the vertices and distances to the sides given by d_a, d_b, and d_c, then PA + PB + PC is at least 2(d_a + d_b + d_c), with equality holding only when the triangle is equilateral and P is its center. The applet provides an interactive geometric construction that visually illustrates the classic proof of this inequality. Through dynamic manipulation, users can drag the point P anywhere within the triangle and observe in real time how the relationship between vertex distances and side distances always satisfies the stated bound. The construction employs several elegant geometric techniques, including rotation and reflection transformations, which are central to the proof strategy. By engaging with this applet, students gain deep insight into how distance relationships behave within a triangle and how geometric transformations such as rotations and reflections can be used as powerful tools in proving non-trivial inequalities. The dynamic nature of the construction allows learners to test the inequality across a wide variety of triangle shapes and point positions, reinforcing their understanding through experimentation. Additionally, the visual proof approach helps students connect algebraic statements with their underlying geometric meaning, making the Erdos-Mordell Inequality more accessible and memorable. This applet is particularly valuable for advanced geometry courses, competition mathematics preparation, or any setting where a deeper appreciation of classical Euclidean results is desired.