Two equilateral triangles share a vertex. Prove that their centres and the highlighted vertex are collinear.
The wall socket

Two equilateral triangles share a vertex. Prove that their centres and the highlighted vertex are collinear.
Two squares of which three vertices form an isosceles triangle. What is the angle α?
A circle and a triangle with two sides tangent to the circle and a side connecting the tangency points. The line segment of the circle centre to the triangle apex is shown. Prove that α = β.
An equilateral triangle with given distances between its vertices and an exterior point. What’s its side length x?
Two arbitrarily placed congruent equilateral triangles. Out of all the possible rotations of the plane, for how many will the image of ABC be A’B’C’?
A square divided by three line segments. What’s the proportion yellow : purple?
A square containing two overlapping equilateral triangles. What’s the length proportion a : b?
A triangle with an inscribed triangle. What fraction of the former does the latter cover in terms of lengths a, b, c, d, e and f?
Two congruent triangles are placed as shown. A line segment passes through the centroid of the yellow triangle. What’s the proportion a : b?
A right triangle with a red and yellow square attached. Two circular arcs centered in the triangle vertices. What is the area of the orange rectangle in terms of red and yellow?