Let be an triangle. On side of the triangle let be two points so that . Let and . Let be the midpoints of the segments: . Proof that can be the three sides of a triangle. Find a condition so that .
With out any loss of generality we consider an oblique co-ordinate system with the origin as A with B and C along x and y axis such that BC=a, AC=b, AB=c the lengths of the given triangle. Now say AD = AB, BE = AB where are some constants. Using the condition we have .
Using Basic Proportionality Theorem in this triangle and the formula for internal division of a line segment according to a given ratio. We can determine the co-ordinates of the points as
Use the Cosine Rule to get the inclination between the co-ordinate axes say as .Now use the distance formula in the oblique system to determine the lengths and show that the sum of lengths of any two sides is greater than the third side so they can form a triangle. will prove to be helpful.
Use the slope equation to get the condition between for .