QL-P24: Intersection QL-P1.QL-P8 ^ QL-P13.QL-P17

QL-P24 is the point of intersection of the lines QL-P1.QL-P8 and QL-P13.QL-P17.

It is also one of the points of intersection of the circles QL-Ci1 (Circumcircle QL-Diagonal Triangle) and QL-Ci6 (Dimidium Circle).

1^st CT-coordinate

(l – m) (l – n) m n

* (T_mn (a⁴ (4 (l – m) (l – n) (l m – l n – m n) (l m – l n + m n) + (l m + l n – m n))

+ b⁴ (m – n) (+l m – l n + m n) / (l – n)

+ c⁴ (m – n) (-l m + l n + m n) / (m – l) – 2 b² c² (m – n)²)

– 2 a² b² ((l – m) (m – n) T_nl – (l m – l n + m n) T_lm)

– 2 a² c² ((l – n) (m – n) T_lm – (l m – l n – m n) T_nl))

where:

T_mn = (l² m² – 2 l² m n + 2 l m² n + l² n² + 2 l m n² – 3 m² n²)

T_lm = (l² m² + 2 l² m n – 2 l m² n – 3 l² n² + 2 l m n² + m² n²)

T_nl = (3 l² m² – 2 l² m n – 2 l m² n – l² n² + 2 l m n² – m² n²)

1^st DT-coordinate

(a² (l²-n²)+b² (m²-n²)) (a² (l²-m²)+c² (-m²+n²))

Properties

• QL-P24 lies on these lines:
  • QL-P1.QL-P8
  • QL-P13.QL-P17
• QL-P24 is the projection of QL-P16 on the line QL-P1.QL-P8.
• QL-P24 is one of the points of intersection of the circles QL-Ci1 (Circumcircle QL-Diagonal Triangle) and QL-Ci6 (Dimidium Circle). The other point of intersection is QL-P17 (QL-Adjunct Quasi Circumcenter).
• QL-P24 = QL-Tf1 (intersection point of the 3 QL-versions of QA-P41.QG-P18).

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