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 week, 4 days 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???

Aaghaz Mahajan - 1 week, 3 days ago

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\(C\), \(E\) and \(H\) are respectively the midpoints of \(AB\), \(AC\), \(DG\).

Thành Đạt Lê - 1 week, 3 days ago

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

Aaghaz Mahajan - 1 week, 3 days ago

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

Aaghaz Mahajan - 1 week, 3 days ago

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@Aaghaz Mahajan Yup, it is.

Thành Đạt Lê - 1 week, 3 days ago

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

Aaghaz Mahajan - 1 week, 3 days ago

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@Aaghaz Mahajan You can prove that.

Thành Đạt Lê - 1 week, 3 days ago

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

Aaghaz Mahajan - 1 week, 3 days ago

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@Thành Đạt Lê 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???

Aaghaz Mahajan - 1 week, 3 days ago

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@Aaghaz Mahajan 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.

Thành Đạt Lê - 1 week, 3 days ago

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