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FEIJERBACH'S THEOREM. The nine-point circle is tangent to the incircle and the three excircles. The points of tangency are called the Feuerhach points. The center of the nine-point circle lies on the Huler line (see Paragraph 3.1. 1-3). | 136 Analytic Geometry i.e. if their direction vectors Ri and R2 are collinear. If the straight lines are given by canonical equations then the condition that they are parallel can be written as Î1 m rn I2 m2 n2 4.6.3.2 Remark. If parallel lines have a common point i.e. ri r2 in parametric equations then they coincide. Example 2. Let us show that the lines x - 1 y - 3 z 2 and x-3 y 1 z are parallel to each other. Indeed condition 4.6.3.2 is satisfied 2 _ 1 _ 2 4 2 4 and hence the lines are parallel. 4.6.3-3. Conditions for two lines to be perpendicular. Two straight lines given by vector parametric equation r r1 tR1 and r r2 ÎR2 are perpendicular if Ri R2 0. 4.6.3.3 If the lines are given by canonical equations then the condition that they are perpendicular can be written as 1112 m1m2 n1n2 0 4.6.3.3a which coincides with formula 4.6.3.3 written in coordinate form. Example 3. Let us show that the lines x - 1 y - 3 z ----- ------ - and 2 1 2 are perpendicular. Indeed condition 4.6.3.3a is satisfied x - 2 y 1 z 1 2 -2 2 1 1 2 2 -2 0 and hence the lines are perpendicular. 4.6.3-4. Theorem on the arrangement of two lines in space. Theorem on the arrangement of two lines in space. Two lines in space can a be skew b lie in the same plane and not meet each other i.e. be parallel c meet at a point d coincide. A general characteristic of all four cases is the determinant of the matrix x2 - x1 l1 I2 y2- y1 m1 m2 Z2 - Z1 n1 n2 4.6.3.4 4.6. Line and Plane in Space 137 whose entries are taken from the canonical equations x - xi y - yi z - zi x - X2 y - y2 z - Z2 ------ -------- --------- and ---------- -------- --------- li mi ni I2 m2 n2 of the lines. In cases a-d of the theorem for the matrix 4.6.3.4 we have respectively a the determinant is nonzero b the last two rows are proportional to each other but are not proportional to the first row c the last two rows are not proportional and the first row is their linear combination d all rows are proportional. 4.6.3-5. Angles between
