GC-03.06 · Ercole Suppa’s Square Midpoint Inradii Ratio (2015)

Discoverer(s): Ercole Suppa | Date: 2015 | Field: \(\mathbb{Q}(\phi)\) | Loci: 4 (Lemoine \(S = 12\))

In square ABCD, let M be the midpoint of side AD. The inradii \(R\) and \(r\) of triangles \(\triangle BCM\) (incircle cEH) and \(\triangle CDM\) (incircle cFP) satisfy \(R / r = \phi\), and segment EF joining their incenters is divided by CM at K with \(|EK|/|KF| = \phi\).
GC-03.06 — Ercole Suppa's Square Midpoint Inradii Ratio (2015) cEH: (y - 1)**2 + (x - 5/2 + sqrt(5)/2)**2 - (-1/2 + sqrt(5)/2)**2 = 0 cFP: (x - 3/2 + sqrt(5)/2)**2 + (y - 3/2 + sqrt(5)/2)**2 - (3/2 - sqrt(5)/2)**2 = 0 CM: -x - 2*y + 2 = 0 EF: x*(-1/2 + sqrt(5)/2) - y - 3*sqrt(5)/2 + 7/2 = 0 Golden major EK = sqrt(5 - 2*sqrt(5)) Golden minor KF = sqrt(2)*sqrt(25 - 11*sqrt(5))/2 B = (2, 2) A = (0, 2) M = (0, 1) H = (2, 1) P = (3/2 - sqrt(5)/2, 0) D = (0, 0) C = (2, 0) E = (5/2 - sqrt(5)/2, 1) F = (3/2 - sqrt(5)/2, 3/2 - sqrt(5)/2) K = (3 - sqrt(5), -1/2 + sqrt(5)/2) B A M H P D C E F K

Given & Constructive Steps

Given: Square ABCD of side \(2\) with D \(= (0,0)\), C \(= (2,0)\), B \(= (2,2)\), A \(= (0,2)\) and midpoint M \(= (0,1)\).

step Join CM and BM, partitioning the lower and right regions into right \(\triangle CDM\) and isosceles \(\triangle BCM\).
step Inscribe circle cEH in \(\triangle BCM\) (incenter E, tangent to BC at H) and circle cFP in \(\triangle CDM\) (incenter F, tangent to CD at P).
step Join incenters EF to intersect shared side CM at K.

Exact Symbolic Proof & Certificate

For \(\triangle BCM\), area \(= 2\) and semiperimeter \(s_1 = 1 + \sqrt{5}\) give inradius \(R = \frac{2}{\sqrt{5}+1} = \frac{\sqrt{5}-1}{2} = 1/\phi\). For right \(\triangle CDM\) with legs \(1, 2\) and hypotenuse \(\sqrt{5}\), inradius \(r = \frac{1 + 2 - \sqrt{5}}{2} = \frac{3 - \sqrt{5}}{2} = 1/\phi^2\). Since E and F lie at perpendicular distances \(R\) and \(r\) on opposite sides of shared line CM, similar right triangles yield \(|EK|/|KF| = R/r = \phi\).

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

EK \(/\) KF \(= \frac{1}{2} + \frac{\sqrt{5}}{2}\) on collinear triple (E, K, F)

Exact Segment Lengths

\(|EK| = \sqrt{5 - 2 \sqrt{5}}\), \(|KF| = \frac{\sqrt{2} \sqrt{25 - 11 \sqrt{5}}}{2}\)

Rational Trigonometry Quadrances

\(Q(E, K) = 5 - 2 \sqrt{5}\), \(Q(K, F) = \frac{25}{2} - \frac{11 \sqrt{5}}{2}\), \(Q_{\text{long}}/Q_{\text{short}} = \frac{\sqrt{5}}{2} + \frac{3}{2} = \phi^2\)

Algebraic Number Field

\(\mathbb{Q}(\phi)\)

Géométrographie Loci

2 given points · 2 lines · 2 circles · 1 polygons (elementary loci \(= 4\), Lemoine \(S = 12\))

Automated Divine Scan

geometor.divine.golden detected 2 exact collinear golden section(s) and 0 connected golden chain(s) in this model:

  • Section (M, K, C): long \(|KC|^2 = \frac{15}{2} - \frac{5 \sqrt{5}}{2}\) vs. short \(|MK|^2 = \frac{35}{2} - \frac{15 \sqrt{5}}{2}\)

  • Section (F, K, E): long \(|KE|^2 = 5 - 2 \sqrt{5}\) vs. short \(|FK|^2 = \frac{25}{2} - \frac{11 \sqrt{5}}{2}\)

Historical Commentary & Attribution

Suppa’s theorem exhibits consecutive negative powers of \(\phi\) (\(\phi^{-1}\) and \(\phi^{-2}\), whose sum \(\phi^{-1} + \phi^{-2} = 1\) equals half the side of the square) as exact Euclidean inradii.