SuccessiveArcs

Successive Arcs in triangles

Consider a triangle t = (ABC). Start with an arbitrary point D on side AC and draw arcs, centered at the vertices and end-points on adjacent sides, so that the start-point of the next is the end-point of the previous. The construction results in an arc-polygon p1 = (DEGIKM), closing back to the start point G (arc-polygon-1).
In the same vein, one constructs the arc-polygon p2 = PNO of the contact points with the incircle of the triangle t. The two arc polygons are indeed closed, their vertices on the same side define equal segments: {DP = EN = OG = PI = ... }, and the vertices of the first arc-polygon p1 are on a circle, concentric to the incircle of t. The triangle t2 = QRS, formed by the diagonals of the arc-polygon p1 is anti-homothetic to the triangle t3 = NOP, formed by the contact points of the triangle t with its incircle. The center of homothety is the Gergonne point U of t. Look in SuccessiveArcsHex.html for a similar theorem on circumscriptible hexagons and suggestions for the proofs.

The proposition can be generalized to circumscribed polygons with an arbitrary odd number, n = 2m+1, of sides. The corresponding procedure of successive arcs produces a 2n-sided, inscribed in circle arc-polygon, whose circumcircle is concentric with the incircle of the original n-gon. An example can be viewed in SuccessiveArcsPenta.html .

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