In Triangle ABC, medians BE and CD are produced respectively to points X and Y such that CD=DX and BE=EY as shown in the figure. Show that the points X, A and Y are collinear. Also show that A is the mid point of XY
Yoyo
To show that the points X, A, and Y are collinear, we need to demonstrate that the line passing through X and Y also passes through A.
Let's start by considering triangle ACD. Since CD = DX, we have a triangle with two sides equal, which means the angles opposite those sides must also be equal. Therefore, angle DCA = angle XDA (1).
Now let's focus on triangle ABE. Similarly, since BE = EY, we have another triangle with two sides equal. Again, the angles opposite those sides must be equal, so angle EBA = angle AEY (2).
Now, let's consider the whole triangle ABC. The points X and Y are the intersections of the medians with the extended sides, which means they divide the medians into equal parts. Therefore, AE = EX and AD = DY.
By combining equations (1) and (2), we can see that angle DCA = angle EBA = angle XDA = angle AEY. This tells us that triangle AEC is similar to triangle DXA. Since the corresponding angles are equal, the sides will be proportional.
Now, let's apply the similarity of triangles AEC and DXA. We have AE/EX = AC/CD and AD/DY = AC/CD, which can be simplified to AE/AC = EX/CD and AD/AC = DY/CD.
Since AE = EX and AD = DY, we can rewrite the above equations as EX/AC = 1 and DY/AC = 1. This implies that EX = AC and DY = AC.
Now, consider the line passing through X and Y. Since EX = AC and DY = AC, we can conclude that EX + EY = AC + AC = 2AC. But we also know that AE = EX and AD = DY, so AE + AD = EX + DY = 2AC. This implies that A is the midpoint of XY.
Hence, we have shown that the points X, A, and Y are collinear and that A is the midpoint of XY.