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Nice Geometry Problem

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Edited by Richard Walker, Thursday, 6 Apr 2023, 18:04

Suppose we take a rectangle and erect equilateral triangles on two of its side, as shown in Fig. 1. Show that pointsC, E and B are the vertices of an equilateral triangle.

Here is my proof: If we draw in the sides of the triangle (Fig. 2) we can see triangles DCE, BEF and DFA all have

a side of length | and one of length ||
an angle of 150 degrees included between those sides.

Consequently the three triangles are all congruent (the same as on another) and DE = EF = FD, so DEF is equilateral as required.
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A triangle problem

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Given an equilateral triangle LMN, let X lie on ML extended, and Y lie on MN, such that LX = NY.

Show that the point P where XY and LN intersect is the midpoint of XY.

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Samurai Puzzle - A Square in a Triangle

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Edited by Richard Walker, Friday, 2 Oct 2020, 09:23

This is a Sangaku-like puzzle (see https://learn1.open.ac.uk/mod/oublog/viewpost.php?post=230691)

We have a square inscribed in a triangle of known base b and height h, as shown. What is the length s of the square's side?

I called this a Samurai puzzle because many of the original sangaku were the work of Samurai.

(Solution in Comments.)

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