Homework is given to a child who is not ready for a math competition...

$A$ and $B$ are fixed points. Determine the locus of point $I$ as $D$ moves on the perpendicular from $A$ to the segment $AB$. Please help me!

Note by Thành Đạt Lê
1 year, 8 months ago

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Are the points C and E also fixed???? Because, if they are, then how do you determine the position of G??? And is H the midpoint of DG???

- 1 year, 8 months ago

$C$, $E$ and $H$ are respectively the midpoints of $AB$, $AC$, $DG$.

- 1 year, 8 months ago

Okay.....but how do we define G???

- 1 year, 8 months ago

Is it the intersection of perpendicular bisector of DC and the line perpendicular to DB passing through E???

- 1 year, 8 months ago

Yup, it is.

- 1 year, 8 months ago

But, I think the method I am using is lengthy...........What I am doing is, I have set up the Coordinate plane with the origin at C, then points A and B are (-2a,0) and (2a,0) and so, the point E becomes (-a,0)..........Now, we can give D the co-ordinates (-2a, 2y).........(Here, a is a real positive number, and y ranges.....).....Now, after some messy equation solving, I have the co-ordinates of G and H..........Here I have another doubt........Is I lying on the circle with DG as a diameter???

- 1 year, 8 months ago

Try to solve the problem. I already know the answer is a combination of half-parabola and a semicircle. But I don't know anything more specific. And $DG$ is a diameter.

- 1 year, 8 months ago

Ohhkay........coz, the picture was a bit misleading.....I assumed that DG is always parallel to AB, which isn't the case........

- 1 year, 8 months ago

You can prove that.

- 1 year, 8 months ago

Umm nope..........that isn't happening always........see my comment and my approach....you'll see what I mean

- 1 year, 8 months ago