Can an Ellipse Help Solve the Scalene Triangle Problem?

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In summary, if C remains constant but the point c moves, the lengths of A and B will change. The locus of c will be an ellipse with the two focuses at A and B, and the vertical line through c will bisect the angle between the blue lines at the point where the tangent to the ellipse is horizontal. This can be seen in the diagram attached. However, there is no purely geometric way to construct this point. In physics, this point is important as it is where light rays from a source at one focus of an ellipse will be focused at the other focus.
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Samwise-zambeezi
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Hi guys!

I've got a problem with a triangle, and I'm frazzled my brain trying to work it out (not even sure if what I'm looking for is possible with the info I have!).

Pic of the offending triangle attached.

Basically, I know the length of C and the sum of lengths A and B.

Now, if C was static and remained unchanged, but if the point c was to move, lengths A and B would change. I'm trying to find the individual lengths of A and B at the point at which, C is split with a vertical into two equal angles (f and g).

Is this possible? Hope this is clear enough, any tips would be a massive help.

Best regards

Swise
 

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If the point $c$ moves so that the sum of its lengths to $a$ and $b$ is constant, then its locus will be an ellipse (the red curve in the diagram) with one focus at $a$ and the other one at $b$. The vertical line through $c$ will bisect the angle between the blue lines through $c$ at the point where the tangent to the ellipse is horizontal, as in the diagram. In other words, this happens at the lowest point of the ellipse. As far as I know, there is no purely geometric way of constructing this point.

In physics, if there is a light source at one focus of an ellipse then the light rays from it will all be focused at the other focus. (That is the reason for using the term focus for these points.) This means that the two blue lines from $c$, to $a$ and $b$, make equal angles with the tangent at $c$. Therefore the line perpendicular to the tangent (in this case, the vertical line) bisects the angle between the blue lines.
 

1. What is a scalene triangle?

A scalene triangle is a type of triangle where all three sides have different lengths. This means that all three angles are also different from each other.

2. How can you determine the area of a scalene triangle?

The area of a scalene triangle can be calculated using the formula: A = 1/2 * base * height, where the base and height are two of the sides of the triangle. If the base and height are not given, you can use other methods such as Heron's formula or trigonometric functions to determine the area.

3. Can a scalene triangle have a right angle?

Yes, a scalene triangle can have a right angle. This type of triangle is called a right scalene triangle. It is possible for a scalene triangle to have any type of angle, including acute and obtuse angles.

4. What is the difference between a scalene triangle and an isosceles triangle?

A scalene triangle has all three sides and angles different from each other, while an isosceles triangle has two sides and two angles that are equal. In an isosceles triangle, the two equal sides are called the legs, and the third side is called the base.

5. How many lines of symmetry does a scalene triangle have?

A scalene triangle has no lines of symmetry. This means that it cannot be folded in half or have any other line of reflection that divides it into two equal parts. However, it can have rotational symmetry, where it can be rotated around a point and still look the same.

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