Interesting question: trajectory of the third point when two triangle vertices slide on a straight line

, The two straight lines intersect at point O. △Abc Vertex a is on one of the straight lines, and vertex B is on the other line. If we keep the edges of △abc unchanged and let vertices A and B slide in the straight line, what kind of figure is the trajectory depicted by vertices C?

 
 
 
 
 
 
 
 
 
 
 
 

Answer: it is an elliptic.

 

To prove this, we create a circle at O, A, and B and record the center as M. Take a straight line over M and C, and the straight line and circle intersect at p and q. Note that because PQ is the diameter of the circle, ∠ poq is always at the right angle. In the process of △abc moving, the diameter of the circle AB/sin (∠ AOB) will remain unchanged. Since the circle diameter is always the same, we can re-describe this circle as a circle with a specified diameter after two points A and B, in this case, the entire circle and the position of p and q are uniquely determined by △abc. In this way, although the location of the arc AP and the arc AQ is constantly changing, its radians remain unchanged, so its circular angle does not change, that is, ∠ AOP and ∠ AOQ are always fixed values. Since the trajectory of a is a straight line, the trajectory of p and q is also a straight line. ∠ Poq is always 90 °, so the trajectory of p and q is actually two vertical lines over the O point.

 

Using these two vertical lines as the coordinate system, we can describe the C trajectory from a new angle. We can regard the PQ of the fixed-line segment as moving on this coordinate system, while the trajectory of point C is the moving trajectory of a certain point on the PQ. Set the distance from C to P to A, and the distance from C to Q to B, because the sum of squares of sin (∠ opq) and sin (∠ oqp) is 1, therefore, the coordinates (x, y) of point C always meet the requirements of x ^ 2/A ^ 2 + y ^ 2/B ^ 2 = 1. Therefore, the trajectory of point C is an elliptic with the op and OQ axes.

Source: http://www.cut-the-knot.org/Curriculum/Geometry/EllipseByVanSchooten.shtml