# Euclid Book 1 Proposition 3

Proposition 3 looks simple, but it uses Proposition 2 which uses Proposition 1. Prop 3 is in turn used by many other Propositions through the entire work.

For debugging it was handy to have a consistent not random pair of given lines, so I made a definite parameter start procedure, selected to look similar to the traditional start points.

```Sub prime_pr3()
'given two unequal lines AB and GH

Dim ptA(0 To 2) As Double
Dim ptB(0 To 2) As Double
Dim ptG(0 To 2) As Double
Dim ptH(0 To 2) As Double

Dim Ax As Double, Ay As Double
Dim Bx As Double, By As Double
Dim Gx As Double, Gy As Double
Dim Hx As Double, Hy As Double

Ax = rnddbl(0, 5)
Ay = rnddbl(0, 5)
Bx = rnddbl(15, 25)
By = rnddbl(0, 10)
Hx = rnddbl(-5, 0)
Hy = rnddbl(12, 18)
Gx = rnddbl(5, 5)
Gy = rnddbl(12, 18)

Call pt(ptA, Ax, Ay, 0)
Call pt(ptB, Bx, By, 0)
Call pt(ptG, Gx, Gy, 0)
Call pt(ptH, Hx, Hy, 0)

Call pr3_sub(ptA, ptB, ptG, ptH)

End Sub

Sub pump_pr3()
'hardwiring two unequal lines AB and GH

Dim ptA(0 To 2) As Double
Dim ptB(0 To 2) As Double
Dim ptG(0 To 2) As Double
Dim ptH(0 To 2) As Double

Call pt(ptA, 1, 1, 0)
Call pt(ptB, 10, 2, 0)
Call pt(ptG, 3, 8, 0)
Call pt(ptH, -3, 9, 0)

Call pr3_sub(ptA, ptB, ptG, ptH)

End Sub

Sub pr3_sub(ptA() As Double, ptB() As Double, ptG() As Double, ptH() As Double)

Dim r As Double
Dim intpts As Variant

Dim ptD(0 To 2) As Double
Dim ptE(0 To 2) As Double

'i copied sub for prop2 and only added object deletes
'at the bottom otherwise this is same as pr2
'pr2 calls its own pr1
Call pr3_pr2_sub(ptA, ptG, ptH)

'vertex found
ptD(0) = ptG1(0)
ptD(1) = ptG1(1)
ptD(2) = ptG1(2)

r = distance(ptA, ptD)

intpts = lineAB.IntersectWith(circF, acExtendNone)
Call intpts_eval(intpts)

'should only be one
ptE(0) = intpts(0)
ptE(1) = intpts(1)
ptE(2) = intpts(2)

lineAB.Delete
'circF.Delete

'labels
Dim th As Double
th = 1#

Call txt_h("A", ptA, th)
Call txt_h("B", ptB, th)
Call txt_h("D", ptD, th)
Call txt_h("E", ptE, th)

End Sub
```