Converses of Napoleon's Theorem

The Development of Napoleon’s Theorem on the Quadrilateral in Case of Outside Direction

“…Look at ∆GQT and ∆BAT on figure 4, ∠TGQ = ∠TBA and ∠GTQ = ∠BTA, so∠GQT = ∠BAT = 90°. • with trigonometric approach is by using the cosine rule and sine rule [6]. So that at ∆ MAN using the cosine rule [5] .…”

“…Furthermore on each equilateral triangle one can obtain a midpoint that is angle from a new equilateral triangle [2]. The new equilateral triangle can be called as Napoleon's triangle [6].…”

“…Many authors have provided a wide range of alternative proofs as well as developed the Napoleon's theorem, for example [6] develops on the inner Napoleon's theorem, whereas [14] describes it if an isosceles (right angle) of mother triangle is given, what properties must the external and internal Napoleon triangles have? What properties must the external and internal Napoleon's triangles of a mother have to ensure that the mother is an isosceles triangle (right angle) triangle?…”

Generalizing the Petr-Douglas-Neumann Theorem onN-gons

“…Using (4) with u = i, v = i + t + 1, and w = j simplifies the second set of terms, reducing (26) to (27) for 0 ≤ i, j ≤ n − 1, with t + 2 terms including a term of the type Z j j if there is one. This is the same as (22), with t + 1 replacing t, so the formula for Q nt extends from t to t + 1.…”

The notable configuration of inscribed equilateral triangles in a triangle

“…Among the most famous classic configurations are the Euler and Simson lines, the Gergonne, Lemoine, Brocard and Nagel concurrences, the Feuerbach, Brocard, Lemoine, Tuker and Taylor circumferences, etc. More modern configurations can be found in the Soddy circles [9], the Euler-Gergonne-Soddy triangle [8], the Torricelli configuration [10], etc. In fact, one can see in [3] four hundred configurations (triangle centers), in [4] even eight hundred configurations are picked up.…”