Draw AR, PS and BT perpendicular to the x-axis. Draw AQ and PC parallel to the x-axis. Then, by the AA similarity criterion, \[ \Delta PAQ \sim \Delta BPC \] Therefore, \[ \frac{PA}{BP} = \frac{AQ}{PC} = \frac{PQ}{BC} \] Now, \[ AQ = RS = OS - OR = x - x_1 \] \[ PC = ST = OT - OS = x_2 - x \] \[ PQ = PS - QS = PS - AR = y - y_1 \] \[ BC = BT - CT = BT - PS = y_2 - y \] Substituting these values in (1), we get \[ \frac{m_1}{m_2} = \frac{x - x_1}{x_2 - x} = \frac{y - y_1}{y_2 - y} \] Taking \[ \frac{m_1}{m_2} = \frac{x - x_1}{x_2 - x}, \] we get \[ x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2} \] Similarly, taking \[ \frac{m_1}{m_2} = \frac{y - y_1}{y_2 - y}, \] we get \[ y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \] So, the coordinates of the point P(x, y) which divides the line segment joining the points A(x_1, y_1) and B(x_2, y_2), internally, in the ratio m_1 : m_2 are \[ \left( \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \right) \]