Equal Areas in Equilateral Triangle
What is it about?
A Mathematical Droodle

19 November 2015, Created with GeoGebra

Explanation

|Activities| |Contact| |Front page| |Contents| |Geometry| |Store|

Copyright © 1996-2017 Alexander Bogomolny

The applet purports to illustrate one of the many properties of an equilateral triangle.

Let M be a point inside an equilateral triangle ABC, with pedal points A', B', and C', as shown. The lines joining M to the six points A, B, C, A', B', C' split the triangle into six smaller ones. The applet suggests that

(1)

Area(AMB') + Area(BMC') + Area(CMA') = Area(A'MB) + Area(B'MC) + Area(C'MA).

(I am grateful to Manuel Silva from Portugal for bringing this problem to my attention.)

Equal Areas in Equilateral Triangle

As in a sister problem, Draw AbAc||BC, BaBc||AC, and CaCb||AB. The three lines split ΔABC into three parallelograms and three smaller equilateral triangles. The three parallelograms are divided by the diagonals joining M to the vertices A, B, C into equal parts, whereas the three equilateral triangles are divided into equal parts by the lines joining M to the pedal points A', B', C'. So, for a proof, one only needs to rearrange the areas in (1).


Related material
Read more...

  • A problem with equilateral triangles: an interactive illustration
  • Six Incircles in Equilateral Triangle
  • Equilateral Triangle, Straight Line and Tangent Circles
  • Equilateral Triangles on Diagonals of Antiequilic Quadrilateral
  • Equilateral Triangles On Sides of a Parallelogram
  • Equilateral Triangles On Sides of a Parallelogram II
  • Equilateral Triangles on Sides of a Parallelogram III
  • Triangle Classification
  • Equilateral Triangles and Incircles in a Square
  • A Circle Rolling in an Equilateral Triangle
  • Viviani's Theorem
  • Regular Polygons in a Triangular Grid
  • Common Centroids Lead to Equilateral Triangle
  • |Activities| |Contact| |Front page| |Contents| |Geometry| |Store|

    Copyright © 1996-2017 Alexander Bogomolny

     62084412

    Search by google: