X(1314)-X(1315) 3rd & 4th Moses intersection
3rd & 4th Moses intersection
The 3rd & 4th Moses intersection are the intersections of the Euler line with the incircle.
The baristic coordinates of X1314, the third Moses intersection are P: (p1 : p2 : p3) with
p1 = [d2 + (4r - R)R + sqrt(Q)]SBSC + (d2 + 2r2 - R2)a2SA
p2 = [d2 + (4r - R)R + sqrt(Q)]SCSA + (d2 + 2r2 - R2)b2SB
p2 = [d2 + (4r - R)R + sqrt(Q)]SASB + (d2 + 2r2 - R2)c2SC
in which:
Q = 4d2R(4r - R) + [d2 - 3R2 + 4r(r + R)]2,
d = distance between the circumcenter X(3) and the orthocenter X(4),
R = circumradius and r = inradius,
SA = (b2 + c2 - a2)/2, and SB and SC are defined cyclically
The baristic coordinates of X1315, the fourth Moses intersection are P: (p1 : p2 : p3) with
p1 = [d2 + (4r - R)R - sqrt(Q)]SBSC + (d2 + 2r2 - R2)a2SA
p2 = [d2 + (4r - R)R - sqrt(Q)]SCSA + (d2 + 2r2 - R2)b2SB
p2 = [d2 + (4r - R)R - sqrt(Q)]SASB + (d2 + 2r2 - R2)c2SC
3de en 4de snijpunt van Moses
Het 3rd & 4th snijpunt van Moses zijn de snijpunten van de rechte van Euler met de ingeschreven cirkel.
De baristische coördinaten van X1314, het derde snijpunt van Moses zijn P: (p1 : p2 : p3) met
p1 = [d2 + (4r - R)R + sqrt(Q)]SBSC + (d2 + 2r2 - R2)a2SA
p2 = [d2 + (4r - R)R + sqrt(Q)]SCSA + (d2 + 2r2 - R2)b2SB
p2 = [d2 + (4r - R)R + sqrt(Q)]SASB + (d2 + 2r2 - R2)c2SC
waarin:
Q = 4d2R(4r - R) + [d2 - 3R2 + 4r(r + R)]2,
d = afstand tussen het zwaartepunt X(3) en het hoogtepunt X(4),
R = straal van omgeschreven cirkel and r = straal van ingeschreven cirkel,
SA = (b2 + c2 - a2)/2 en SB en SC bereken je analoog
De baristische coördinaten van X1315, het vierde snijpunt van Moses zijn P: (p1 : p2 : p3) met
p1 = [d2 + (4r - R)R - sqrt(Q)]SBSC + (d2 + 2r2 - R2)a2SA
p2 = [d2 + (4r - R)R - sqrt(Q)]SCSA + (d2 + 2r2 - R2)b2SB
p2 = [d2 + (4r - R)R - sqrt(Q)]SASB + (d2 + 2r2 - R2)c2SC