Harmonic sequence point - Chowareten

Japanese: 調和列点 - ちょうわれってん

When the line segment AB is internally and externally divided by points C and D in the same ratio, the four points A, B, C, and D are said to be harmonic sequence points. In this case, C and D are also said to divide A and B harmoniously, and one of C and D is called the harmonic conjugate point of the other with respect to A and B. If A, B, C, and D are harmonic sequence points, then B, A, D, C and C, D, A, and B are also harmonic sequence points. When there are three points A, B, and C on a line l , take point P outside l and connect P with A, B, and C, then take point Q on PC, and let I be the intersection point of AQ and BP, J be the intersection point of BQ and AP, and D be the intersection point of IJ and l . Then D is the harmonic conjugate point of C with respect to A and B (Figure 1).

Source: Heibonsha World Encyclopedia, 2nd Edition Information

Japanese:

点C,Dにより線分ABが同じ比に内分および外分されているとき，4点A,B,C,Dは調和列点であるという。このとき，また，C,DはA,Bを調和に分かつともいい，CとDの一方をA,Bに関する他方の調和共役点という。A,B,C,Dが調和列点ならば，B,A,D,CおよびC,D,A,Bも調和列点である。1直線l上に3点A,B,Cがあるとき，点Pをlの外にとってPとA,B,Cを結び，次に点QをPC上にとって，AQとBPの交点をI,BQとAPの交点をJとし，IJとlとの交点をDとすれば，DはA,Bに関するCの調和共役点である(図1)。

出典　株式会社平凡社世界大百科事典第２版について　情報

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