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.
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.
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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