Iterations in Geometry, a generalization

Iterations that start with a point in the plane of ΔABC and move first half way to vertex A, and from there half way to vertex B, and then half way to vertex C, and so on, converge to a triangle defined by three points

(1) (A + 2B + 4C)/7, (B + 2C + 4A)/7, (C + 2A + 4B)/7.

This process can be generalized in two ways. First, on every step, the distance to a subsequent vertex could be cut in a ratio Rn:Rd different from 1:1. It is easy to see that if

α = Rn:Rd + 1

then the iterations converge to

(2) (A + αB + α²C)/σ, (B + αC + α²A)/σ, (C + αA + α²B)/σ.

where σ = 1 + α + α². If D is the first of the three points in (2), then the cevian fromA through D, cuts BC in the ratio α:1.

Secondly, the process can apply to any N-gon, not necessarily a triangle. (2) has a natural generalization for N vertices. In the applet below, three numbers N, Rn, and Rd are all modifiable. Clicking a little off but to the right of the number's vertical center line will increase the number, clicking to the left will decrease it.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

What if applet does not run?

Limits in Geometry

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