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Geometry

Section Formula

Let $$P$$ be the internal division point of the line segment $$AB$$ joining points $$A=(-9,6)$$ and $$B=(12,4)$$ in the ratio $$t:(1-t)$$ for $$t>0$$. If $$P$$ lies in the first quadrant of the Cartesian plane, what is the range of the real number $$t?$$

Consider the line segment joining points $$A=(-6,-7)$$ and $$B=(8,-1)$$. If $$P$$ is the intersection point of this line segment and the $$y\text{-axis},$$ and $$P$$ internally divides the line segment in the ratio $$m:n,$$ where $$m$$ and $$n$$ are coprime positive integers, what is the value of $$mn?$$

Let $$P$$ be the internal division point of $$\overline{AB}$$ joining points $$A$$ and $$B$$ in the ratio $$7:8,$$ and $$Q$$ be the external division point of $$\overline{AB}$$ in the ratio $$8:7.$$ If the length of $$\overline{PQ}$$ is $$\lvert\overline{PQ}\rvert=113,$$ what is $$\lvert\overline{AB}\rvert?$$

Let $$R$$ be the external division point of the line segment $$PQ$$ joining points $$P=(-1,1)$$ and $$Q=(11,16)$$ in the ratio $$k:5.$$ If point $$R$$ (the sought-after external division point) is on the line $$x+y = -18,$$ then what is the positive constant $$k?$$

The line segment $$AB$$ connecting two points $$A=(23, 14)$$ and $$B=(2, 5)$$ on the xy-plane is internally divided $$2:1$$ by a point $$P=(x, y$$). What is $$x+y$$?

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