geometry advanced

[paul.350]

Hello, Please solve the following question.
1. Construct a right triangle with given hypotenuse c such that
the median drawn to the hypotenuse is the geometric mean of
the two legs of the triangle.

2. An arbitrary point M is selected in the interior of the segment AB. The squares AMCD and MBEF are constructed on the same side of AB, with the segments AM and MB as their respective bases. The circles circumscribed about these squares, with centers P and Q, intersect at M and also at another point N. Let N’ denote the point of intersection of the straight lines AF and BC.
(i) Prove that the points N and N’ coincide.
(ii) Prove that the straight lines MN passes through a fixed point S independent of the Choice of M.
(iii) Find the locus of the mid points of the segments PQ as M varies between A and B.

Thanks in advance
-paul

 

[Denis]

Please read: "Read Before Posting!!"

 

[soroban]

Hello, Paul!

I think I've solved 2(iii)

2. An arbitrary point \(\displaystyle M\) is selected in the interior of the segment \(\displaystyle AB.\)
The squares \(\displaystyle AMCD\) and \(\displaystyle MBEF\) are constructed on the same side of \(\displaystyle AB\), with the segments \(\displaystyle AM\) and \(\displaystyle MB\) as their respective bases.
The circles circumscribed about these squares, with centers \(\displaystyle P\) and \(\displaystyle Q\), intersect at \(\displaystyle M\) and also at another point \(\displaystyle N.\)
Let \(\displaystyle N'\) denote the point of intersection of the straight lines \(\displaystyle AF\) and \(\displaystyle BC.\)

(iii) Find the locus of the midpoint \(\displaystyle X\) of the segment \(\displaystyle PQ\) as \(\displaystyle M\) varies between \(\displaystyle A\) and \(\displaystyle B.\)

  D                 C
  *-----------------*
  |                 |
  |                 |
  |        P        |F        E
  |        o        *---------*
  |        |      o |    Q    |
  |        |a/2   X |    o    |
  |        |        |    |b/2 |
  *--------+--------*----+----*
  A        a        M    b    B
  : - - - - - -  c  - - - - - :

\(\displaystyle \text{The squares are }AMCD\text{ and }MBEF\text{ with centers }P\text{ and }Q.\)

\(\displaystyle \text{Let }c = AB,\,\text{ a constant.}\)

\(\displaystyle \text{Let }a = AM,\;b = MB,\;\text{ where }a+b\,=\,c.\)


\(\displaystyle \text{Place vertex }A\text{ at the origin.}\)

\(\displaystyle P\text{ has coordinates: }\:\left(\frac{a}{2},\:\frac{a}{2}\right)\)

\(\displaystyle Q\text{ has coordinates: }\:\left(a+\frac{b}{2},\:\frac{b}{2}\right)\)


\(\displaystyle X \text{ is the midpoint of }PQ\text{ and has coordinates:}\)

. . \(\displaystyle x \:=\:\frac{\frac{a}{2} + (a+\frac{b}{2})}{2} \:=\:\frac{3a+b}{4}\)

. . \(\displaystyle y\:=\:\frac{\frac{a}{2} + \frac{b}{2}}{2} \:=\:\frac{a+b}{4}\)


\(\displaystyle \text{Since }a+b\,=\,c,\,\text{ we have: }\)

. . \(\displaystyle x \:=\:\frac{2a + (a+b)}{4} \:=\:\frac{2a+c}{4} \:=\:\frac{a}{2} + \frac{c}{4}\)

. . \(\displaystyle y \:=\:\frac{a+b}{4} \:=\:\frac{c}{4}\)



\(\displaystyle \text{We see that the "height" of }X\text{ is a constant.}\)

\(\displaystyle \text{and the }x\text{-coordinate is a linear function: }\:x \;=\;\frac{1}{2}a + \frac{c}{4}\)

. . \(\displaystyle \text{Its domain is: }\:[0,\:c]\)

. . \(\displaystyle \text{Its range is: }\:\left[\tfrac{1}{4}c,\:\tfrac{3}{4}c\right]\)