Proving Euler Line Parallel to Side BC in Triangle ABC

In summary, the conversation discusses proving the concurrency of AY, BZ, and CX in a triangle ABC where equilateral triangles XAB, YBC, and ZAC are formed on it. Additionally, it suggests proving that the tangent of angles B and C is equal to 3 by showing that the Euler line is parallel to side BC and that Z=X=Y at the orthocentre. The concept of the Euler line, which connects the orthocentre, circumcentre, ninepoint circlecentre, and centroid, is also mentioned.
  • #1
vaishakh
334
0
Can anyone help me solve the following question. ABC is any triangle. XAB, YBC and ZAC are equilateral triangles formed on this triangle. Prove that AY, BZ and CX are concurrent.


In a triangle ABC, the Euler line is parallel to side BC. Prove that tanB*tanC = 3.

I just need a hint. I don't know how to start with the second problem?
 
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  • #2
For the first, try to proove that Z=X=Y at the orthocentre.
For the second, I have no clue what a Euler line is so I can't help there.
 
  • #3
Euler line is on joning the orthocentre, circumcentre, ninepoint circlecentre and the centroid.
 

1. What is the Euler Line in a triangle?

The Euler Line is a line that passes through the centroid, circumcenter, and orthocenter of a triangle. It is also known as the "nine-point line" because it passes through nine important points of a triangle.

2. How is the Euler Line parallel to the side BC in triangle ABC?

In order to show that the Euler Line is parallel to the side BC in triangle ABC, we must first prove that the centroid, circumcenter, and orthocenter of the triangle are collinear. This can be done by using the fact that the centroid divides each median into a 2:1 ratio. Once we have established this, we can then use the fact that the circumcenter is equidistant from the three vertices of the triangle and the orthocenter is the intersection of the altitudes. From this, we can conclude that the Euler Line is indeed parallel to the side BC.

3. What are the properties of the Euler Line?

The Euler Line has several properties, including:

  • It passes through the centroid, circumcenter, and orthocenter of a triangle.
  • It is parallel to the triangle's shortest side.
  • It is perpendicular to the triangle's longest side.
  • It is the center of the nine-point circle.
  • It divides the triangle's perimeter into two equal parts.

4. How is the Euler Line related to the nine-point circle?

The nine-point circle is a circle that passes through the nine important points of a triangle, including the endpoints of the Euler Line. In fact, the Euler Line is the diameter of the nine-point circle, with the midpoint of the line being the center of the circle.

5. Can the Euler Line be proven using different methods?

Yes, there are several different ways to prove that the Euler Line is parallel to a side of a triangle. Some methods include using vector geometry, complex numbers, or trigonometry. Each method may have its own advantages and may be more suitable for different situations.

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