Thabit: Triangles Built on Intersecting Circles
Equilateral, unequal and two-equal-sided triangles from overlapping equal circles
The page is headed with the name Thabit and carries four diagrams of two equal, overlapping circles whose circumferences pass through each other's centres. On the first, an equilateral triangle stands on the line joining the centres a and b, with the intersections lettered c and S, and the equal sides are derived from the definition of the circle (ab = ac, ba = bc). Further figures construct a triangle "of three unequal" sides (points n, h, r, m, t, f, o) and one with "two sides equal" (r, S, t, m, c, b), reasoning that the radii ct and cS are equal because they run from centre to circumference.
On this page
Equilateral triangle on two equal circles
By the first postulate, circle cbS is drawn about point a and circle caS about point b; by the definition of the circle, ab equals ac and ba equals bc, so the triangle on ab is equilateral. The intersections are lettered c above and S below.
Two circles sharing a diameter
A second view shows the two overlapping equal circles with their diameters coincident, the centres lettered a and b and the intersections of the diameter with the circles lettered l and f.
A triangle of three unequal sides
In the next figure the larger circles meet at point n, and a scalene triangle is built with vertices among the points h, r, m, t, f and o — captioned "of three unequal" sides.
A triangle with two sides equal
In the last figure two smaller overlapping circles sit on the common perpendicular; the larger circles meet at t above and m below, and lines from c and b to t give a triangle with "two sides equal", since radii ct and cS run from centre to circumference and are equal.