SOLUTION: Write a coordinate proof for each statement. The three segments joining the midpoints of the sides of an isosceles triangle form another isosceles triangle.

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Question 675193: Write a coordinate proof for each statement. The three segments joining the midpoints of the sides of an isosceles triangle form another isosceles triangle.
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
If the black triangle below is isosceles, and the vertices of the green
triangle are the midpoints of the sides of the black triangle, that the
green triangle is also isosceled:



Since the black triangle is isosceles we can let it be triangle PQR,
where the vertices are P(0,b), Q(-a,0), and R(a,0), where a and b are
both positive numbers:

 

Now we will let S, T. and U be the midpoints respectively of
PQ, PR and QR.

We will use the midpoint formula to find the coordinates of S,T, and U:

For S:
Midpoint = 
Midpoint of PQ = S%28matrix%281%2C3%2C++++++%28-a%2B0%29%2F2%2C+++%22%2C%22%2C+%280%2Bb%29%2F2%29%29
Midpoint of PQ = S%28matrix%281%2C3%2C++++++-a%2F2%2C+++%22%2C%22%2C+b%2F2%29%29

For T:
Midpoint = 
Midpoint of PR = T%28matrix%281%2C3%2C++++++%280%2Ba%29%2F2%2C+++%22%2C%22%2C+%28b%2B0%29%2F2%29%29
Midpoint of PR = T%28matrix%281%2C3%2C++++++a%2F2%2C+++%22%2C%22%2C+b%2F2%29%29

For U:
Midpoint = 
Midpoint of QR = U%28matrix%281%2C3%2C++++++%28-a%2Ba%29%2F2%2C+++%22%2C%22%2C+%280%2B0%29%2F2%29%29
Midpoint of QR = U(0,0)   



Now all we need is to show that US = UT.

We will use the distance tance formula to show that:

 d = sqrt%28%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2%29
SU = sqrt%28%280-%28-a%2F2%29%29%5E2%2B%280-b%2F2%29%5E2%29
SU = sqrt%28%28a%2F2%29%5E2%2B%28-b%2F2%29%5E2%29
SU = sqrt%28a%5E2%2F2%5E2%2Bb%5E2%2F2%5E2%29
SU = sqrt%28a%5E2%2F4%2Bb%5E2%2F4%29
SU = sqrt%28%28a%5E2%2Bb%5E2%29%2F4%29
SU = sqrt%28a%5E2%2Bb%5E2%29%2F2

 d = sqrt%28%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2%29
TU = sqrt%28%28a%2F2-0%29%5E2%2B%28b%2F2-0%29%5E2%29
TU = sqrt%28%28a%2F2%29%5E2%2B%28b%2F2%29%5E2%29
TU = sqrt%28a%5E2%2F2%5E2%2Bb%5E2%2F2%5E2%29
TU = sqrt%28a%5E2%2F4%2Bb%5E2%2F4%29
TU = sqrt%28%28a%5E2%2Bb%5E2%29%2F4%29
TU = sqrt%28a%5E2%2Bb%5E2%29%2F2

So SU = TU and therefore triangle STU is isosceles.

Edwin