+-11 A right isosceles triangle is a triangle with two sides that are equal in length and are perpendicular to each other. The medial triangle of triangle ABC is the triangle whose vertices are the midpoints of the sides of triangle ABC. Prove that the medial triangle of a right isosceles triangle is also a right isosceles triangle.
+129852 
Let triangle ABC be an isoscelses right triangle with equal legs AC and BC and hypotenuse BA
Since E and D split sides AC and AB proportionally (they are midpoints), then ED is parallel to BC
And for the same reason DF is parallel to AC and EF is parallel to AB
And angle AED = angle ACB = 90
And since EF is parallel to AB, then angle EFC = angle ABC
But EF is a transversal cutting parallel segments , so angle EFC = angle FED
So angle FED also equals angle ABC
And since Dfis parallel to CA, then angle DFB = angle ACB = 90
And angle DBF = angle ABC
So, by AA congruency, triangle DFB is similar to triangle ACB
And angle FDB = angle CAB
But DF is a transversal cutting parallels ED and CB......
Then angle EFD = angle FDB
But FDB = angle CAB
So angle CAB = angle EFD
So angle CAB = EFD
And angle ABC = angle FED
But CB = AC so angles CAB and ABC are equal
So angle EFD and FED are equal
So triangle EFD is similar to triangle CAB by AA congruency
And since ABC is an isosceles right triangle, median triangle FDE is also an isosceles right triangle
-11 wow! I never thought I would see THE CPhill in my questions. Anyways, thanks so much for the detailed help!
