1. A method of operating apparatus for guiding insertion of a needle of a syringe into a blood vessel in a body such that part of the needle is inserted into the body and part of the needle remains outside the body, the apparatus including: a planar camera and a lateral camera, each of which includes a built-in near infrared camera and a visible light camera and which are so placed as to face the blood vessel and to be parallel to each other; a monitor, and a computer, the method characterized by comprising the steps of: capturing images of the blood vessel, the syringe, and the part of the needle of the syringe remaining outside the body with the planar camera and with the lateral camera, images of the part of the needle inserted into the body being unable to be captured by the visible light cameras of the planar camera and the lateral camera, and images of the blood vessel being captured only by the near infrared cameras of the planar camera and the lateral camera; obtaining a planar image forming an X-Y coordinate plane by overlaying the images of the blood vessel, the syringe, and the needle of the syringe; obtaining a lateral image forming an X-Z coordinate plane by overlaying the images of the blood vessel, the syringe, and the needle of the syringe, the lateral image having a common X-axis with the planar image; displaying the planar image and the lateral image on the monitor; using the computer to detect the image of the needle of the syringe from each of the planar image and the lateral image displayed on the monitor, and to perform a process of displaying, on a line extending from each image of the needle of the syringe, an image of the part of the needle inserted into the body and a straight extended line representing advancing of the needle, by: using the computer to detect, in the lateral image, an X-coordinate value of an intersection at which the extended line of the needle of the syringe and the blood vessel intersects each other; using the computer to read the X-coordinate value of the intersection on the X-axis; and using the computer to detect, from the planar image, a point which has the same value as the X-coordinate value of the intersection and which is present on the blood vessel, and to perform a process of displaying the point on the monitor as an insertion target, wherein the step of detecting the X-coordinate value, the step of reading the X-coordinate value, and the step of performing the process of displaying the insertion target are performed repeatedly.