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Orthocenter

 

Orthocenters are nothing but the midpoint of all the perpendiculars drawn from each vertex of a triangle. Ortho is the Greek word for ‘right’, the point where all the altitudes of any triangle meet are termed as the orthocenter of that triangle. An orthocenter is very important in studying the different characteristics of a triangle concerning its other sides. The altitudes drawn from one vertex to the other side (opposite to that vertex) eventually meet at some point in or outside the triangle. This point of convergence is the orthocenter of a triangle.

For an orthocenter to exist, there must be three main components, a triangle, an altitude, and a vertex. We all know that a triangle is a three-sided figure having three vertices. A vertex is known as the point where two lines meet and extend an angle (acute, obtuse, or right angle). An altitude is a line that passes from one vertex to the side opposite of the vertex and may or may not bisect the triangle.

For a better understanding of what orthocenters are, look at the diagram below. Below is a triangle QPR of undefined dimensions; let us make three altitudes PM, QN, and RO, and the point where all these altitudes meet be A. This point-A is known as the orthocenter.

Different types of triangles have different types of properties of the orthocenter. As mentioned above, the orthocenter may lie inside a triangle or may lie outside of it. Different types of triangles affect the position of the orthocenter. For example, in an equilateral triangle (equal-sided triangle), the orthocenter and centroid are the same things. Let us look at more properties of the orthocenter according to the form of the triangle.

  • In an acute angle triangle, the orthocenter is always inside a triangle.
  • In an obtuse angle triangle, the orthocenter is always outside a triangle.
  • In a right-angled triangle, the orthocenter lies on the vertex from where the right angle is made.
  • The orthocentre divides the altitudes into two different sections. The product of these sections results in being equal to the original length of the perpendiculars.

Formula for Orthocenter of a Triangle:

Let us assume an acute angle triangle, having vertices ABC and coordinates are A (a1, b1), B (a2, b2), and C (a3, b3), respectively. Let the perpendiculars drawn from each vertex meet at a point O having coordinates (a, b). O is the orthocenter of this triangle. To find the value of O, first, we need to determine the slope of each side of the triangle.

Applying the formula of slope on two sides keeping ‘A’ as a common vertex

Slope of AB = b2 – b1 / a2 – a1

Slope of AC= b3 – b1 / a3 – a1

Next, we need to find the slope of the altitudes of the triangle ABC 

We know that, Perpendicular slope of line = −1 / slope of the line

Therefore, the slope of the respective altitudes

Slope of CM = -1 / slope of AB

The slope of BL = −1 / slope of AC

Generalizing these equations, we get = 

Slope of CM = b – b1/ a – a1

Slope of BL = b – b2 / a – a2

We can use these equations to find the coordinates of the orthocenter of any triangle.

 

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