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.

jeffthememeguy Nov 20, 2020

#1**+2 **

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

CPhill Nov 20, 2020

#2**0 **

wow! I never thought I would see THE CPhill in my questions. Anyways, thanks so much for the detailed help!

jeffthememeguy
Nov 20, 2020