Orthocenter in a Triangle | Equation of Locus

Here you will learn formula for orthocenter in a triangle, ex-centers and equation of locus of a point.

Let’s begin –

Orthocenter in a Triangle

It is the point of intersection of perpendiculars drawn from the vertices on the opposite sides of a triangle and it can be obtained by solving the equation of any two altitudes.

Note :

(i)  If the triangle is right angled, the orthocenter is the point where right angle is formed.

(ii)  Co-ordinates of circumcenter is (\(x_1tanA+x_2tanB+x_3tanC\over {tanA+tanB+tanC}\),\(y_1tanA+y_2tanB+y_3tanC\over {tanA+tanB+tanC}\))

Ex-centers

The center o a circle which touches side BC and the extended portions of sides AB and AC is called the ex-center of triangle ABC with respect to the vertex A. It is denoted by \(I_1\) and its coordinates are

\(I_1\) (\(-ax_1+bx_2+cx_3\over {-a+b+c}\),\(-ay_1+by_2+cy_3\over {-a+b+c}\))

Similarly ex-centers of triangle ABC with respect to vertices B and C are denoted by \(I_2\) and \(I_3\) respectively, and

\(I_2\) (\(ax_1-bx_2+cx_3\over {a-b+c}\),\(ay_1-by_2+cy_3\over {a-b+c}\)), \(I_3\) (\(ax_1+bx_2-cx_3\over {a+b-c}\),\(ay_1+by_2-cy_3\over {a+b-c}\))

Locus – Equation of locus

The locus of a moving point is the path traced out by that point under one or more geometrical conditions

(a) Equation of Locus :

The equation to a locus is the relation which exists between the coordinates of any point on the path, and which holds for no other point except those lying on the path.

(b) Procedure for finding the equation of the locus of a point :

(i)  If we are finding the equation of the locus of a point P, assign coordinates (h,k) to P.

(ii)  Express the given condition as equations in terms of the known quantities to facilitate calculations. We sometimes include some unknown quantities known as parameters.

(iii)  Eliminate the parameters, so that the eliminant contains only h,k and known quantities.

(iv) Replace h by x, and k by y, in the eliminant. The resulting equation would be the equation of the locus of P.

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