Fixed Point in Isosceles and Equilateral Triangles

The applet below illustrates a construction problem suggested by an anonymous visitor:

Take an variable isosceles triangle ABD on base AB (fixed) with D free to move on the perpendicular bisector of AB. On AD and BD place externally equilateral triangles ADF and BDE. Finally take C on the perpendicular bisector of AB such that DFCE is a rhombus.

As D moves on the perpendicular bisector of AB, the linkage should be free to flex. However, it will be observed that C does not move: ΔABC is always equilateral.



This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Explanation

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Copyright © 1996-2017 Alexander Bogomolny

Fixed Point in Isosceles and Equilateral Triangle

Take an variable isosceles triangle ABD on base AB (fixed) with D free to move on the perpendicular bisector of AB. On AD and BD place externally equilateral triangles ADF and BDE. Finally take C on the perpendicular bisector of AB such that DFCE is a rhombus.

As D moves on the perpendicular bisector of AB, the linkage should be free to flex. However, it will be observed that C does not move: ΔABC is always equilateral.



This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Solution

The problem is just a reformulation of the construction of an equilateral triangle with a rusty compass.

It is not necessary for the equilateral triangles to be constructed externally. The statement remains valid if both are constructed either externally or internally.


Related material
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  • Equilateral and 3-4-5 Triangles
  • Rusty Compass Construction of Equilateral Triangle
  • Equilateral Triangle on Parallel Lines
  • Equilateral Triangle on Parallel Lines II
  • When a Triangle is Equilateral?
  • Viviani's Theorem
  • Viviani's Theorem (PWW)
  • Tony Foster's Proof of Viviani's Theorem
  • Viviani in Isosceles Triangle
  • Viviani by Vectors
  • Slanted Viviani
  • Slanted Viviani, PWW
  • Morley's Miracle
  • Triangle Classification
  • Napoleon's Theorem
  • Sum of Squares in Equilateral Triangle
  • A Property of Equiangular Polygons
  • Parallels through the Vertices of Equilateral Triangle
  • |Activities| |Contact| |Front page| |Contents| |Geometry|

    Copyright © 1996-2017 Alexander Bogomolny

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