X₄₈₃

RADICAL CENTER OF AJIMA-MALFATTI CIRCLES

The Ajima-Malfatti circles are described at X₁₇₉. (Peter Yff, 6/1/98)

5, 6, 7 3, 4, 5 Kepler Kimberling

X₄₈₃ (RADICAL CENTER OF AJIMA-MALFATTI CIRCLES) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {8, 178}, {174, 175}

• Related Centers: X₁ (INCENTER), X₈ (NAGEL POINT), X₁₇₈ (2nd MID-ARC POINT), X₁₇₄ (YFF CENTER OF CONGRUENCE), X₁₇₅ (ISOPERIMETRIC POINT)

Trilinears: \(\sec^{2} A/4 : \sec^{2} B/4 : \sec^{2} C/4\)

Barycentrics: \(\sin A \sec^{2} A/4 : \sin B \sec^{2} B/4 : \sin C \sec^{2} C/4\)

Search Value (6, 9, 13): \(1.470496841714\)

▸ Alternate Coordinates1 form

• Trilinears: \(1/(1 + \cos A/2) : 1/(1 + \cos B/2) : 1/(1 + \cos C/2)\)

▸ Incident Central Lines3 lines

{8, 178}, {173, 180}, {174, 175}

▸ Functional Relations2 relations

• X₄₈₃ = X₃₇₂-of-BCI-triangle

• X₄₈₃ = BCI-isogonal conjugate of X₁

Notes

Let (I_a) be the incircle of BCI, where I=X₁, and define (I_b) and (I_c) cyclically. Let E_a be the touchpoint of (I_a) and BC, and define E_b and E_c cyclically. Let S_a be the insimilicenter of (I_b) and (I_c), and define S_b and S_c cyclically. The lines E_aS_a, E_bS_b, E_cS_c concur in X₄₈₃. Let L_a be the internal tangent, other than AI, of (I_b) and (I_c). Define L_b and L_c cyclically. Then L_a, L_b, L_c concur in X₄₈₃. (Randy Hutson, January 29, 2018)