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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 http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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What if applet does not run?

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Copyright © 1996-2012 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 http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

Buy this 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 Read more...

	Equilateral and 3-4-5 Triangles
	Rusty Compass Construction of Equilateral Triangle
	Equilateral Triangle on Parallel Lines
	When a Triangle is Equilateral?
	Viviani's Theorem
	Viviani's Theorem (PWW)
	Viviani in Isosceles Triangle
	Viviani by Vectors
	Morley's Miracle
	Triangle Classification
	Napoleon's Theorem
	Sum of Squares in Equilateral Triangle
	A Property of Equiangular Polygons

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Copyright © 1996-2012 Alexander Bogomolny

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