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Geometric Proof of Perpendicular Segments in Polygons

Cincinnati's proof of an equilateral triangle and any interior point, where perpendicular segments from the point to any sides sum will be constant. Explore various polygons like Rectangle, Rhombus, Regular Pentagon, and Hexagon. Understand the consistency in distances between parallel lines and equal triangles areas.

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Geometric Proof of Perpendicular Segments in Polygons

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  1. Cincinnati’s Proof Given an equilateral triangle and any interior point, the perpendicular segments from the point to any sides sum will be constant.

  2. First step of proof: divide the original triangle into three smaller triangles using the perpendicular segments as their altitudes.

  3. What about other polygons: Rectangle Rhombus Regular Pentagon Hexagon with only opposite sides parallel

  4. Rectangle - Yes • This is because the distance between parallel lines is constant. Distance is always measured with a perpendicular.

  5. Rhombus • Prove using: • A. Distance between parallels is constant. • B. Area of triangles as an equal length that can be factored.

  6. Regular Pentagon • Proof by area of triangles

  7. Hexagon with only opposite sides parallel • Proof by distance between parallel lines are constant.

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