answer:Let's denote the common difference of the arithmetic sequence {a_n} as d. Since S_3=12 and a_2+a_4=4, we have: - 3a_1+3d=12 - 2a_1+4d=4 Solving these equations, we find a_1=6 and d=-2. Therefore, the sum of the first 6 terms is S_6=6times6+ frac{6times5}{2}times(-2)=6. Hence, the answer is boxed{6}. This problem can be solved by using the general formula for the nth term and the sum formula of an arithmetic sequence. It tests reasoning and computational skills and is considered a medium-difficulty problem.

answer:1. Let's analyze the pencil of lines through the point (C) intersected by the parallel lines (AA_1) and (MN). 2. Since (C_1) lies on (AC), and the line through (C_1) is parallel to (AA_1), it implies that (MN) is also parallel to (AA_1). Therefore, we have the relationship: [ frac{MN}{C_1N} = frac{AA_1}{PA_1} = frac{C_1N}{ND} ] 3. By cross-multiplying the last two fractions, we get: [ frac{MN}{C_1N} cdot frac{C_1N}{ND} = frac{AA_1}{PA_1} ] This simplifies to: [ frac{MN}{ND} = frac{AA_1}{PA_1} ] 4. Now, by the basic proportionality (Thales' theorem), since (MN) is parallel to (AA_1), the triangles (C_1NM) and (C_1NAA_1) are similar. 5. Due to similarity of these triangles, we have the following sides proportional: [ frac{MN}{C_1N} = frac{C_1N}{ND} ] 6. Multiplying these proportions gives: [ MN cdot ND = C_1N cdot C_1N ] 7. Hence, this relation simplifies to: [ MN cdot ND = C_1N^2 ] 8. Therefore, we have proved that: [ N C_{1}^2 = M N cdot N D ] # Conclusion: [ boxed{N C_{1}^2 = M N cdot N D} ]

answer:1. **Given Point and Circle Information:** The point ( A ) has coordinates ( (b, 0) ), and point ( P ) is any point on the circle given by the equation ( x^2 + y^2 = 9 ). 2. **Geometric Considerations:** Let ( M ) be a point on line segment ( AP ) such that the ratio ( frac{AM}{MP} = frac{1}{2} ). To find the equation of the locus of point ( M ), observe the following: 3. **Construction and Relationship:** Draw a line ( MN parallel PO ) that intersects the ( x )-axis at point ( N ). By construction, we see that ( |MN| ) is a constant length. Specifically, ( |MN| = 1 ) since ( frac{AM}{MP} = frac{1}{2} ) leads to ( 2AM = MP ), and hence the distance from ( M ) to ( N ) on a line parallel to ( PO ) is consistently 1 unit away. 4. **Fixed Point on ( x )-axis:** The point ( N ) on the ( x )-axis is fixed at ( (4, 0) ). 5. **Circle Definition:** From the plane geometry definition, a point that maintains a constant distance from a fixed point traces out a circle. Thus, the circle's equation is centered at ( N = (4,0) ) with radius equal to ( 1 ). 6. **Equation of the Circle:** Using the standard form of the circle equation: [ (x - 4)^2 + y^2 = 1 ] By verifying these steps, we determine that the locus of point ( M ), as ( P ) moves along the circle ( x^2 + y^2 = 9 ), traces out another circle centered at ( (4,0) ) with radius (1). Therefore, the equation of the locus of ( M ) is: [ boxed{(x-4)^2 + y^2 = 1} ]

answer:**Analysis:** Based on the context and life experience, understanding of mass units, length units, time units, and data sizes, we can determine that Xiaohong's height should be measured in "centimeters"; weight should be measured in "kilograms"; the length of a math book should be measured in "centimeters", the thickness of a math book should be measured in "millimeters", the cargo capacity of a truck should be measured in "tons", the approximate daily sleep time for children should be measured in "hours", and the height of a big tree should be measured in "meters". Therefore, the appropriate units are: - Xiaohong's height is 140 boxed{text{centimeters}}, weight 23 boxed{text{kilograms}}. - A math book is about 20 boxed{text{centimeters}} long, and about 7 boxed{text{millimeters}} thick. - A truck can carry 4 boxed{text{tons}} of cargo. - Children sleep about 9 boxed{text{hours}} a day. - A big tree is about 12 boxed{text{meters}} tall.