FourEulerCircles

Four Euler Circles

Consider a quadrangle A₁A₂A₃A₄ and its four "partial" triangles defined through its diagonals: A₁A₂A₃, A₂A₃A₄, ... etc. The Euler circles of these four triangles have a common point (C).

Indeed, consider the "opposite" Euler circles c₁, c₂ passing through the middles E, F, G and D, B, G of the sides and the diagonal A₁A₃. Measure the angles at their second intersection point C: angle(GCD) = π-angle(GBD) = angle(A₃DB) = angle(A₃GB). Analogously angle(FCG) = angle(A₃GE). Thus angle(FCD) = angle(FA₃D) = angle(FHD) and the third Euler circle c₃, passing through F, H, D passes also through C. Analogously the fourth circle passes also through C.
The theorem is intimately related to the problem of constructing a rectangular hyperbola passing through four non-orthocentric points discussed in the file RectHypeThroughFourPts.html .

Four Euler Circles

See Also