Promoting Mathematical Thinking
  • Promoting Mathematical Thinking ... the home of John Mason & Anne Watson
    • IMP
    • PMThetaAtHome
    • Anne Watson >
      • Anne's PodCasts
      • Publications
      • Concerns
      • ATM triangles
      • Dose of Don
      • ToulouseGrids
    • AW & JM Joint Presentations
    • John Mason >
      • JHM Bio
      • Piranesi Inspired Pictures
      • JHM Presentations
      • Studies
      • Other Projects
      • JHM Publications
      • JHM-Draughty-Drafts
      • Applets >
        • Reasoning Without Arithmetic
        • Number Formats
        • Reasoning about Numbers
        • Reasoning about Graphs of Functions
        • Fractions & Decimals
        • More or Less Grids
        • Calgary Grids
        • Related Polygons
        • Polynomial Studies
        • Polygonal Projections
        • Sundaram Grids
        • Geometrical Applets
        • Structured Variation Grids
        • Animated Situations
        • Outer Inner Area Ratios
        • Clocks
        • Balancing Tasks
        • Linear Algebra

Outer-Inner Polygon Area Ratios

 For a triangle, Routh's Theorem gives the ratio of the areas of the outer to inner triangles formed by lines joining each vertex to points on the opposite edge.

​In the case shown here, in which the sides are divided in the ratio pf 2 : 1 cyclically, the ratio os 7 : 1
Picture

For a quadrilateral (depicted below), in the case of joining each vertex to the midpoint of the next but one edge (all done clockwise or all done counter-clockwise) it is easy to be misled by rounding errors into believing that the ratio of the areas of the outer to inner quadrilaterals is 5. In fact it lies between 5 and 6.
Picture
Picture



​There is an applet for displaying the diagrams for this paper and for assisting in the exploration can be found here.
Picture
Picture
Proudly powered by Weebly