Harmonic Relation and the Euler Segment Using directed distance, the Cross Ratio of four c...

Harmonic Relation and the Euler Segment Using directed distance, the Cross Ratio of four collinear points A, B, C, and D is defined by the number

If that Cross-Ratio has the value -1, then the four points A, B, C, and D are said to be in harmonic relation. An example is provided by taking die four points on the ,x-axis: A:x1 = 0, B: x2 = 15, C: x3 = 12, and D: x4 = 20 (as indicated in the figure). Using the formula for directed distance P1P2 = x2 – x1, we find

AC = 12 – 0 = 12 BD = 20 – 15 = 5

AD = 20 – 0 = 20 BC = 12 – 15 = -3

And

THE HARMONIC CONTRUCTION

(a) Make a copy of the Euler segment from the construction of Problem 13 by using EDIT, Copy, and Paste, after selecting segment and points H, F, G, and R. Translate (by dragging) to a new position on the screen and delete the rest of the figure. (If segment is too small, select point H and use the Select, Dilation tool of Sketchpad to enlarge it; this will not affect any ratios of distances on that segment.) Calculate the Cross-Ratio (HG, RF) = (HR • GF)/(HF • GR). You should obtain 1 for the answer. (The Distance feature on Sketchpad does not recognize directed distance, so the answer will be positive.)

(b) Follow these steps in Sketchpad to obtain an interesting geometric construction, involving the harmonic relation:

[1] Select point S not on line and construct segments .

[2] Locate T, any point on segment . and construct segment . Select the point U of intersection of segments and .

[3] Construct line and locate its intersection, V, with segment . Hide this line, and construct segment .

[4] Construct line .

Did anything happen? Drag points S and T about and observe the effect? State any theorem (or theorems) you think might be true \HRGSTUV is sometimes referred to as the harmonic configuration (or construction) relative to H, R, G, and F].

NOTE: This so-called harmonic configuration is valid for any four collinear points that are in the harmonic relation (Cross-Ratio = -1). The proof of this belongs more appropriately to that area of geometry known as Projective Geometry.