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 . 1-3). | 136 Analytic Geometry . 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 Remark. If parallel lines have a common point . 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 is satisfied 2 _ 1 _ 2 4 2 4 and hence the lines are parallel. . 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. If the lines are given by canonical equations then the condition that they are perpendicular can be written as 1112 m1m2 n1n2 0 which coincides with formula 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 is satisfied x - 2 y 1 z 1 2 -2 2 1 1 2 2 -2 0 and hence the lines are perpendicular. . 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 . 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 . 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 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. . Angles between
