10 Simple Steps to Find Orthocentre in

**Introduction**

The orthocenter of a triangle is the point where the perpendicular bisectors of the three sides intersect. It is a key point in Euclidean geometry, and has many interesting properties. Finding the orthocenter of a triangle is a straightforward process, and can be done using a variety of methods.

One method for finding the orthocenter is to use the fact that it is the point of concurrency of the three altitudes of the triangle. The altitude of a triangle is a line segment drawn from a vertex to the opposite side, perpendicular to that side. To find the orthocenter, first find the altitudes of the triangle by drawing perpendicular lines from the vertices to the opposite sides. The point where these three altitudes intersect is the orthocenter.

Another method for finding the orthocenter is to use the fact that it is the centroid of the triangle formed by the three perpendicular bisectors of the sides. The centroid of a triangle is the point where the three medians of the triangle intersect. A median of a triangle is a line segment drawn from a vertex to the midpoint of the opposite side. To find the orthocenter, first find the perpendicular bisectors of the three sides of the triangle. Then, find the centroid of the triangle formed by these three lines. The centroid is the orthocenter.

Understanding the Concept of Orthocentre

Understanding the orthocentre of a triangle requires a solid grasp of two key concepts: perpendicular bisectors and concurrences. Let’s delve into each of these elements to build a comprehensive foundation.

Perpendicular Bisectors

A perpendicular bisector is a line that intersects a segment at its midpoint and forms a 90-degree angle with the segment. In the context of triangles, each side has a corresponding perpendicular bisector. These lines are particularly useful because they always pass through the triangle’s circumcenter, which is the point where the perpendicular bisectors of all three sides intersect.

To visualize this, imagine drawing the perpendicular bisectors of each side of a triangle on a piece of paper. The point where these lines meet is the triangle’s circumcenter.

The following table summarizes the properties of perpendicular bisectors:

Property	Definition
Bisects the segment	Intersects the segment at its midpoint
Perpendicular to the segment	Forms a 90-degree angle with the segment

Identifying the Perpendicular Bisectors of a Triangle

To locate the orthocentre, the first step is to determine the perpendicular bisectors of the triangle. A perpendicular bisector is a line that passes through the midpoint of a side and is perpendicular to that side. This construction effectively divides the side into two equal segments.

To locate the perpendicular bisector of a given side, follow these steps:

Mark the midpoint of the side using a compass or ruler.
Draw a line segment perpendicular to the side passing through the midpoint.

Repeat this process for the other two sides of the triangle. The perpendicular bisectors will intersect at a single point, which is the orthocentre of the triangle.

Properties of Perpendicular Bisectors:

The perpendicular bisectors of a triangle have several important properties:

Property	Description
Concurrent Point	The perpendicular bisectors intersect at a single point called the orthocentre.
Equal Distances	Each perpendicular bisector divides its respective side into two equal segments.
Altitude	The perpendicular bisector of a base is also an altitude, or the perpendicular from a vertex to the opposite side.
Circumcircle	The perpendicular bisectors are perpendicular to the sides of the triangle and thus tangent to the triangle’s circumcircle.

Verifying the Orthocentre Property

To verify the orthocentre property, we need to show that the three altitudes of a triangle intersect at a single point. Let’s consider a triangle ABC with altitudes AD, BE, and CF.

1. AD is perpendicular to BC

By definition, an altitude of a triangle is a line segment drawn from a vertex to the opposite side perpendicular to that side. Therefore, AD is perpendicular to BC.

2. BE is perpendicular to AC

Similarly, BE is perpendicular to AC.

3. CF is perpendicular to AB

By the same logic, CF is perpendicular to AB.

4. Proving that AD, BE, and CF intersect at a single point

Now, we need to prove that AD, BE, and CF intersect at a single point. To do this, we will use the fact that the altitudes of a triangle intersect at a point known as the orthocentre.

Theorem
In any triangle, the altitudes intersect at a single point, called the orthocentre.

Since AD, BE, and CF are the altitudes of triangle ABC, they must intersect at a single point, which is the orthocentre. This proves the orthocentre property.

Properties of the Orthocentre and Inradius

Definition of Orthocentre

In geometry, the orthocentre of a triangle is the point of intersection of the three altitudes. An altitude is a perpendicular line from a vertex to the opposite side.

Important Properties

The orthocentre of a triangle has several important properties:

It is equidistant from the three vertices.

It is inside the triangle if the triangle is acute, outside if the triangle is obtuse, and on a side if the triangle is right-angled.

It is the centre of the triangle’s nine-point circle, which is a circle that passes through nine specific points associated with the triangle.

Inradius

The inradius of a triangle is the radius of the inscribed circle, which is the largest circle that can be drawn inside the triangle and tangent to all three sides.

Relationship between Orthocentre and Inradius

The orthocentre and inradius of a triangle are related by the following theorem:

The product of the three segments from the orthocentre to the vertices is equal to the square of the inradius.

This theorem can be expressed mathematically as:

Formula of Orthocentre
OA x OB x OC = (r²)

where O is the orthocentre, A, B, and C are the vertices, and r is the inradius.

What is an Orthocentre?

In geometry, the orthocentre of a triangle is the point where the three altitudes (perpendicular lines from each vertex to the opposite side) intersect. It is also known as the point of concurrency for the triangle’s altitudes.

Applications of the Orthocentre in Problem Solving

The orthocentre can be used to solve a variety of geometry problems. Some of the most common applications include:

1. Finding the area of a triangle

The area of a triangle can be calculated using the orthocentre and the lengths of the three sides. The formula for the area is A = (1/2)bh, where b is the length of the base and h is the height, or distance from the base to the orthocentre.

2. Finding the centroid of a triangle

The centroid of a triangle is the point where the three medians (lines connecting each vertex to the midpoint of the opposite side) intersect. The centroid is also located on the line connecting the orthocentre to the midpoint of the longest side of the triangle.

3. Finding the circumcenter of a triangle

The circumcenter of a triangle is the centre of the circle that passes through all three vertices. The circumcenter is located on the line connecting the orthocentre to the midpoint of the longest side of the triangle.

4. Finding the incenter of a triangle

The incenter of a triangle is the centre of the circle that is inscribed in the triangle, meaning it touches all three sides. The incenter is located on the line connecting the orthocentre to the midpoint of the shortest side of the triangle.

5. Finding the orthocentre of a triangle

There are several ways to find the orthocentre of a triangle. One method is to use the following steps:

Draw the three altitudes of the triangle.
Find the point where the three altitudes intersect. This is the orthocentre.

6. Orthocentre and Triangles

Orthocentres can also be used to construct certain types of triangles. For example, if you know the lengths of the three altitudes of a triangle, you can construct the triangle using the following steps:

Draw a line segment with length equal to the length of the longest altitude.
Draw a perpendicular bisector to the line segment at one end.
Draw a perpendicular bisector to the line segment at the other end.
The point of intersection of the two perpendicular bisectors is the orthocentre of the triangle.
Draw the three altitudes from the orthocentre to the sides of the triangle to complete the triangle.

Altitude	Midpoint
h_a	M_a
h_b	M_b
h_c	M_c

Alternative Methods for Determining the Orthocentre

7. Using the Circumradius

If the triangle has a circumcircle, then the orthocentre is the intersection of the perpendicular bisectors of the sides. This method can be used to find the orthocentre even if the triangle is not drawn to scale.

Let’s draw a diagram:

In the diagram, the circumcenter is O and the orthocenter is H.

From the triangle OAH, we have the following right angles:

∠OHA = 90°
∠OAH = 90°

Therefore, OA = OH.

Orthocentres in Special Triangles (Equilateral, Isosceles)

Equilateral Triangles

In an equilateral triangle, all three sides are equal. Therefore, the three altitudes are equal. This means that the orthocentre is the same point as the centroid and circumcentre. It is located at the intersection of the perpendicular bisectors of the three sides.

Isosceles Triangles

In an isosceles triangle, two sides are equal. Therefore, the two altitudes corresponding to those sides are equal. This means that the orthocentre is located on the perpendicular bisector of the third side.

Isosceles Right Triangle

In an isosceles right triangle, the two altitudes corresponding to the legs are equal. This means that the orthocentre is located at the midpoint of the hypotenuse. The orthocentre is also the circumcentre of the triangle.

Triangle Type	Location of Orthocentre
Equilateral Triangle	Circumcentre and centroid
Isosceles Triangle	Perpendicular bisector of the third side
Isosceles Right Triangle	Midpoint of the hypotenuse

The Orthocentre

In the realm of geometry, the orthocentre of a triangle is an intriguing and pivotal point where the altitudes intersect. Altitudes are lines perpendicular to the sides of a triangle, extending from the vertices to the opposite sides. The orthocentre can be thought of as the geometric “centre” of a triangle, even though it may not always reside within the triangle itself.

Altitude

The altitudes of a triangle serve as the foundation for locating the orthocentre. Each altitude is perpendicular to the side of the triangle that it intersects, forming a right angle with that side. In a right triangle, the altitude is also known as the hypotenuse, which is the longest side of the triangle and connects the right angle to the opposite vertex.

Circumcentre

The circumcentre of a triangle is another significant point associated with the orthocentre. It is the centre of the circle that circumscribes the triangle, meaning it passes through all three vertices of the triangle. The circumcentre represents the geometric “centre” of the triangle, regardless of its shape or size.

The Relationship between Orthocentre and Circumcentre

Euler’s Orthocentre-Circumcentre Relationship

A profound relationship exists between the orthocentre and the circumcentre of a triangle. This relationship, known as Euler’s Orthocentre-Circumcentre Relationship, states that the orthocentre, circumcentre, and centroid of a triangle are collinear. The centroid is yet another geometric centre, representing the geometric average of the triangle’s vertices. Euler’s theorem dictates that these three points lie on a straight line, with the circumcentre positioned midway between the orthocentre and the centroid.

This relationship has significant consequences in trigonometry and geometry. It implies that the triangle’s altitudes, circumradius, and the distances from the vertices to the orthocentre are intimately linked. Euler’s relationship provides a powerful tool for solving various geometric problems involving triangles.

Point	Definition
Orthocentre	Intersection point of altitudes
Circumcentre	Centre of the circumscribing circle
Centroid	Average point of vertices

These geometric relationships are fundamental to understanding the properties of triangles and solving a wide range of mathematical problems. They provide insights into the geometry of triangles and their associated properties, offering a deeper appreciation of the beauty and elegance of geometry.

How to Find the Orthocentre

The orthocentre of a triangle is the point where the three altitudes (perpendiculars from the vertices to the opposite sides) of the triangle intersect. Finding the orthocentre can be useful for various geometric constructions and calculations.

Steps to Find the Orthocentre:

Draw the Altitudes: Construct the altitudes from each vertex to the opposite side of the triangle.
Find the Intersections: Locate the points where the altitudes intersect. The point of intersection is the orthocentre.

Example:

Consider a triangle with vertices A(2,3), B(6,1), and C(1,5).

Altitude from A: Perpendicular from A to BC with equation y = (5/4)x – 13/4
Altitude from B: Perpendicular from B to AC with equation y = -4/5x + 26/5
Intersection: Solve the above equations simultaneously to find the orthocentre (12/13, 67/13)