The 3rd & 4th Moses intersection are the intersections of the [url=https://en.wikipedia.org/wiki/Euler_line]Euler line[/url] with the incircle.[br]The baristic coordinates of X[sub]1314[/sub], the third Moses intersection are P: (p[sub]1[/sub] : p[sub]2[/sub] : p[sub]3[/sub]) with[br]p[sub]1[/sub] = [d[sup]2[/sup] + (4r - R)R + sqrt(Q)]S[sub]B[/sub]S[sub]C[/sub] + (d[sup]2[/sup] + 2r[sup]2[/sup] - R[sup]2[/sup])a[sup]2[/sup]S[sub]A[/sub] [br]p[sub]2[/sub] = [d[sup]2[/sup] + (4r - R)R + sqrt(Q)]S[sub]C[/sub]S[sub]A[/sub] + (d[sup]2[/sup] + 2r[sup]2[/sup] - R[sup]2[/sup])b[sup]2[/sup]S[sub]B[/sub] [br]p[sub]2[/sub] = [d[sup]2[/sup] + (4r - R)R + sqrt(Q)]S[sub]A[/sub]S[sub]B[/sub] + (d[sup]2[/sup] + 2r[sup]2[/sup] - R[sup]2[/sup])c[sup]2[/sup]S[sub]C[br][/sub] in which:[br]Q = 4d[sup]2[/sup]R(4r - R) + [d[sup]2[/sup] - 3R[sup]2[/sup] + 4r(r + R)][sup]2[/sup],[br]d = distance between the circumcenter X(3) and the orthocenter X(4),[br]R = circumradius and r = inradius,[br]S[sub]A[/sub] = (b[sup]2[/sup] + c[sup]2[/sup] - a[sup]2[/sup])/2, and S[sub]B[/sub] and S[sub]C[/sub] are defined cyclically[br][br]The baristic coordinates of X[sub]1315[/sub], the fourth Moses intersection are P: (p[sub]1[/sub] : p[sub]2[/sub] : p[sub]3[/sub]) with[br]p[sub]1[/sub] = [d[sup]2[/sup] + (4r - R)R - sqrt(Q)]S[sub]B[/sub]S[sub]C[/sub] + (d[sup]2[/sup] + 2r[sup]2[/sup] - R[sup]2[/sup])a[sup]2[/sup]S[sub]A[/sub] [br]p[sub]2[/sub] = [d[sup]2[/sup] + (4r - R)R - sqrt(Q)]S[sub]C[/sub]S[sub]A[/sub] + (d[sup]2[/sup] + 2r[sup]2[/sup] - R[sup]2[/sup])b[sup]2[/sup]S[sub]B[/sub] [br]p[sub]2[/sub] = [d[sup]2[/sup] + (4r - R)R - sqrt(Q)]S[sub]A[/sub]S[sub]B[/sub] + (d[sup]2[/sup] + 2r[sup]2[/sup] - R[sup]2[/sup])c[sup]2[/sup]S[sub]C[/sub]

Het 3rd & 4th snijpunt van Moses zijn de snijpunten van de [url=https://en.wikipedia.org/wiki/Euler_line]rechte van Euler[/url] met de ingeschreven cirkel.[br]De baristische coördinaten van X[sub]1314[/sub], het derde snijpunt van Moses zijn P: (p[sub]1[/sub] : p[sub]2[/sub] : p[sub]3[/sub]) met[br]p[sub]1[/sub] = [d[sup]2[/sup] + (4r - R)R + sqrt(Q)]S[sub]B[/sub]S[sub]C[/sub] + (d[sup]2[/sup] + 2r[sup]2[/sup] - R[sup]2[/sup])a[sup]2[/sup]S[sub]A[/sub] [br]p[sub]2[/sub] = [d[sup]2[/sup] + (4r - R)R + sqrt(Q)]S[sub]C[/sub]S[sub]A[/sub] + (d[sup]2[/sup] + 2r[sup]2[/sup] - R[sup]2[/sup])b[sup]2[/sup]S[sub]B[/sub] [br]p[sub]2[/sub] = [d[sup]2[/sup] + (4r - R)R + sqrt(Q)]S[sub]A[/sub]S[sub]B[/sub] + (d[sup]2[/sup] + 2r[sup]2[/sup] - R[sup]2[/sup])c[sup]2[/sup]S[sub]C[br][/sub] waarin:[br]Q = 4d[sup]2[/sup]R(4r - R) + [d[sup]2[/sup] - 3R[sup]2[/sup] + 4r(r + R)][sup]2[/sup],[br]d = afstand tussen het zwaartepunt X(3) en het hoogtepunt X(4),[br]R = straal van omgeschreven cirkel and r = straal van ingeschreven cirkel,[br]S[sub]A[/sub] = (b[sup]2[/sup] + c[sup]2[/sup] - a[sup]2[/sup])/2 en S[sub]B[/sub] en S[sub]C[/sub] bereken je analoog[br][br]De baristische coördinaten van X[sub]1315[/sub], het vierde snijpunt van Moses zijn P: (p[sub]1[/sub] : p[sub]2[/sub] : p[sub]3[/sub]) met[br]p[sub]1[/sub] = [d[sup]2[/sup] + (4r - R)R - sqrt(Q)]S[sub]B[/sub]S[sub]C[/sub] + (d[sup]2[/sup] + 2r[sup]2[/sup] - R[sup]2[/sup])a[sup]2[/sup]S[sub]A[/sub] [br]p[sub]2[/sub] = [d[sup]2[/sup] + (4r - R)R - sqrt(Q)]S[sub]C[/sub]S[sub]A[/sub] + (d[sup]2[/sup] + 2r[sup]2[/sup] - R[sup]2[/sup])b[sup]2[/sup]S[sub]B[/sub] [br]p[sub]2[/sub] = [d[sup]2[/sup] + (4r - R)R - sqrt(Q)]S[sub]A[/sub]S[sub]B[/sub] + (d[sup]2[/sup] + 2r[sup]2[/sup] - R[sup]2[/sup])c[sup]2[/sup]S[sub]C[/sub]