calculate-the-harmonic-conjugated-point-of-a-triplet-of-aligned-points

Calculate the Harmonic Conjugated Point of a Triplet of Aligned Points

The image shows how we can obtain the Harmonic Conjugated Point of three aligned points A, B, C.

• We choose any point L, that is not in the line with A, B and C. We form the triangle ABL

• Then we draw a line from point C that intersects the sides of this triangle at points M and N respectively.

• We draw the diagonals of the quadrilateral ABNM; they are AN and BM and they intersect at point K

• We unit, with a line L and K, and this line intersects the line of points A, B and C at point D

The point D is named the Conjugated Harmonic Point of the points A, B, C. You can get more knowledge related with this point at: (https://en.wikipedia.org/wiki/Projective_harmonic_conjugate)

If we apply the theorems of Ceva (https://en.wikipedia.org/wiki/Ceva%27s_theorem) and Menelaus (https://en.wikipedia.org/wiki/Menelaus%27_theorem) we will have this formula:

AC, in the above formula is the length of the segment of points A to C in this direction and its value is:

AC = xA - xC

Transform the above formula using the coordinates xA, xB, xC and xD

The task is to create a function harmon_pointTrip(), that receives three arguments, the coordinates of points xA, xB and xC, with values such that : xA < xB < xC, this function should output the coordinates of point D for each given triplet, such that

xA < xD < xB < xC, or to be clearer

let's see some cases:

harmPoints(xA, xB, xC) ------> xD # the result should be expressed up to four
 decimals (rounded result)
harmPoints(2, 10, 20) -----> 7.1429 # (2 < 7.1429 < 10 < 20, satisfies the constraint)
harmPoints(3, 9, 18) -----> 6.75