# 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.

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Explanation ### 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.

### 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. • 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
• 