A conjecture about the intersections of three hyperboles related to any triangle

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Given any triangle $triangle ABC$, we build the hyperbole with foci in $A$ and $B$ and passing through $C$.

enter image description here

Similarly, we can build other two hyperboles, one with foci in $A$ and $C$ and passing through $B$ (red), and one with foci in $B$ and $C$ and passing through $A$ (green).

enter image description here

The first part of my conjecture is that the three hyperboles always intersect in two points $D$ and $E$.

enter image description here

Moreover, the ellipse with foci in these two points $D$ and $E$, and passing through one of the three vertices of the triangle $triangle ABC$, pass also through the other two vertices.

enter image description here

These are probably obvious results. However, is there an elementary proof for these conjectures?

Thanks for your help! Sorry in case this is too trivial.

edited 3 hours ago

asked 4 hours ago

Andrea Prunotto

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up vote
8
down vote

favorite

Given any triangle $triangle ABC$, we build the hyperbole with foci in $A$ and $B$ and passing through $C$.

enter image description here

Similarly, we can build other two hyperboles, one with foci in $A$ and $C$ and passing through $B$ (red), and one with foci in $B$ and $C$ and passing through $A$ (green).

enter image description here

The first part of my conjecture is that the three hyperboles always intersect in two points $D$ and $E$.

enter image description here

Moreover, the ellipse with foci in these two points $D$ and $E$, and passing through one of the three vertices of the triangle $triangle ABC$, pass also through the other two vertices.

enter image description here

These are probably obvious results. However, is there an elementary proof for these conjectures?

Thanks for your help! Sorry in case this is too trivial.

edited 3 hours ago

asked 4 hours ago

Andrea Prunotto

1,459726

add a commentÂ |Â

up vote
8
down vote

favorite

Given any triangle $triangle ABC$, we build the hyperbole with foci in $A$ and $B$ and passing through $C$.

enter image description here

Similarly, we can build other two hyperboles, one with foci in $A$ and $C$ and passing through $B$ (red), and one with foci in $B$ and $C$ and passing through $A$ (green).

enter image description here

The first part of my conjecture is that the three hyperboles always intersect in two points $D$ and $E$.

enter image description here

Moreover, the ellipse with foci in these two points $D$ and $E$, and passing through one of the three vertices of the triangle $triangle ABC$, pass also through the other two vertices.

enter image description here

These are probably obvious results. However, is there an elementary proof for these conjectures?

Thanks for your help! Sorry in case this is too trivial.

edited 3 hours ago

asked 4 hours ago

Andrea Prunotto

1,459726

Given any triangle $triangle ABC$, we build the hyperbole with foci in $A$ and $B$ and passing through $C$.

enter image description here

Similarly, we can build other two hyperboles, one with foci in $A$ and $C$ and passing through $B$ (red), and one with foci in $B$ and $C$ and passing through $A$ (green).

enter image description here

The first part of my conjecture is that the three hyperboles always intersect in two points $D$ and $E$.

enter image description here

Moreover, the ellipse with foci in these two points $D$ and $E$, and passing through one of the three vertices of the triangle $triangle ABC$, pass also through the other two vertices.

enter image description here

These are probably obvious results. However, is there an elementary proof for these conjectures?

Thanks for your help! Sorry in case this is too trivial.

geometry triangle conic-sections geometric-construction

edited 3 hours ago

asked 4 hours ago

Andrea Prunotto

1,459726

edited 3 hours ago

asked 4 hours ago

Andrea Prunotto

1,459726

edited 3 hours ago

asked 4 hours ago

Andrea Prunotto

1,459726

asked 4 hours ago

Andrea Prunotto

1,459726

asked 4 hours ago

Andrea Prunotto

1,459726

add a commentÂ |Â

2 Answers
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Try this:

Let E be a point of intersection of black and red hyperbole. Then

black hyperbole: $EA-EB=AC-CB$

red hyperbole: $EA-EC=AB-BC$

Subtracting, you get

$$EC-EB=AC-AB$$ therefore green hyperbole also passes through E.

Do the same for point D, saying that D is a point of intersection of red and green hyperbole, and you will get that black hyperbole also passes through D.

edited 3 hours ago

answered 3 hours ago

MrDudulex

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Second part: Let $a=BC$, $b = CA$ and $c =AB$.

We have $$EB-EC = DC-DB = c-b$$
$$colorredEB-EA = DA-DB = a-b$$ $$DA-DC = FC-FA =c-a$$

If $$AD+AE = 2d$$ then since $$BD+BE = (DA+b-a)+(EA+a-b)=2d$$

so $B$ also lies on this ellipse. The same is true for $C$.

answered 3 hours ago

greedoid

31.6k114287

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2 Answers
2

active

oldest

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2 Answers
2

active

oldest

votes

up vote
2
down vote

Try this:

Let E be a point of intersection of black and red hyperbole. Then

black hyperbole: $EA-EB=AC-CB$

red hyperbole: $EA-EC=AB-BC$

Subtracting, you get

$$EC-EB=AC-AB$$ therefore green hyperbole also passes through E.

Do the same for point D, saying that D is a point of intersection of red and green hyperbole, and you will get that black hyperbole also passes through D.

edited 3 hours ago

answered 3 hours ago

MrDudulex

42519

add a commentÂ |Â

up vote
2
down vote

Try this:

Let E be a point of intersection of black and red hyperbole. Then

black hyperbole: $EA-EB=AC-CB$

red hyperbole: $EA-EC=AB-BC$

Subtracting, you get

$$EC-EB=AC-AB$$ therefore green hyperbole also passes through E.

Do the same for point D, saying that D is a point of intersection of red and green hyperbole, and you will get that black hyperbole also passes through D.

edited 3 hours ago

answered 3 hours ago

MrDudulex

42519

add a commentÂ |Â

up vote
2
down vote

Try this:

Let E be a point of intersection of black and red hyperbole. Then

black hyperbole: $EA-EB=AC-CB$

red hyperbole: $EA-EC=AB-BC$

Subtracting, you get

$$EC-EB=AC-AB$$ therefore green hyperbole also passes through E.

Do the same for point D, saying that D is a point of intersection of red and green hyperbole, and you will get that black hyperbole also passes through D.

edited 3 hours ago

answered 3 hours ago

MrDudulex

42519

Try this:

Let E be a point of intersection of black and red hyperbole. Then

black hyperbole: $EA-EB=AC-CB$

red hyperbole: $EA-EC=AB-BC$

Subtracting, you get

$$EC-EB=AC-AB$$ therefore green hyperbole also passes through E.

Do the same for point D, saying that D is a point of intersection of red and green hyperbole, and you will get that black hyperbole also passes through D.

edited 3 hours ago

answered 3 hours ago

MrDudulex

42519

edited 3 hours ago

answered 3 hours ago

MrDudulex

42519

answered 3 hours ago

MrDudulex

42519

answered 3 hours ago

MrDudulex

42519

add a commentÂ |Â

up vote
2
down vote

Second part: Let $a=BC$, $b = CA$ and $c =AB$.

We have $$EB-EC = DC-DB = c-b$$
$$colorredEB-EA = DA-DB = a-b$$ $$DA-DC = FC-FA =c-a$$

If $$AD+AE = 2d$$ then since $$BD+BE = (DA+b-a)+(EA+a-b)=2d$$

so $B$ also lies on this ellipse. The same is true for $C$.

answered 3 hours ago

greedoid

31.6k114287

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up vote
2
down vote

Second part: Let $a=BC$, $b = CA$ and $c =AB$.

We have $$EB-EC = DC-DB = c-b$$
$$colorredEB-EA = DA-DB = a-b$$ $$DA-DC = FC-FA =c-a$$

If $$AD+AE = 2d$$ then since $$BD+BE = (DA+b-a)+(EA+a-b)=2d$$

so $B$ also lies on this ellipse. The same is true for $C$.

answered 3 hours ago

greedoid

31.6k114287

add a commentÂ |Â

up vote
2
down vote

Second part: Let $a=BC$, $b = CA$ and $c =AB$.

We have $$EB-EC = DC-DB = c-b$$
$$colorredEB-EA = DA-DB = a-b$$ $$DA-DC = FC-FA =c-a$$

If $$AD+AE = 2d$$ then since $$BD+BE = (DA+b-a)+(EA+a-b)=2d$$

so $B$ also lies on this ellipse. The same is true for $C$.

answered 3 hours ago

greedoid

31.6k114287

Second part: Let $a=BC$, $b = CA$ and $c =AB$.

We have $$EB-EC = DC-DB = c-b$$
$$colorredEB-EA = DA-DB = a-b$$ $$DA-DC = FC-FA =c-a$$

If $$AD+AE = 2d$$ then since $$BD+BE = (DA+b-a)+(EA+a-b)=2d$$

so $B$ also lies on this ellipse. The same is true for $C$.

answered 3 hours ago

greedoid

31.6k114287

answered 3 hours ago

greedoid

31.6k114287

answered 3 hours ago

greedoid

31.6k114287

answered 3 hours ago

greedoid

31.6k114287

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