X(999): "X(999) is the radical center of the mixtilinear incircles."
伪旁切圆
与AB、AC延长线相切且与外接圆外切
与AB、AC的切点连线中点为A-旁心
X(55)
X(6244): "X(6244) = radical center of mixtilinear excircles"
伪外接圆
过B、C且与内切圆内切
与内切圆的切点在连接A-旁心JA和内切圆在BC的切点D的直线上；过JAD的中点
X(479): "Let A′ be the point in which the incircle is tangent to a circle that passes through vertices B and C, and define B and C cyclically. The lines AA′, BB′, CC′ concur in X(479)."
X(57): "Let Oa be the circle passing through B and C, and tangent to the incircle. Define Ob and Oc cyclically. Let A′ be the point of tangency of Oa and the incircle, and define B′ and C′ cyclically. Triangle A′B′C′ is perspective to the intouch triangle at X(57). Also, X(57) is the radical center of circles Oa, Ob, Oc."
伪旁接圆
过B、C且与A-旁切圆内切
与A-旁切圆的切点在连接内心I和A-旁切圆在BC的切点D′的直线上；过ID′的中点
X(5423): "Let A′ be the point in which the A-excircle is tangent to the circle OA that passes through vertices B and C, and define B′ and C′ cyclically. The lines AA′, BB′, CC′ concur in X(5423)."
