**Midpoint of Line Segment Joining the Centroid and Orthocenter of a Triangle**

To find the midpoint of the line segment joining the centroid and orthocenter of a triangle with vertices (a, b), (a, c), and (d, c), we need to follow a step-by-step process. Let's go through each step in detail:

**Step 1: Find the Centroid**
The centroid of a triangle is the point of intersection of its medians. A median is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. To find the centroid, we can use the following formula:

- The x-coordinate of the centroid (Cx) is the average of the x-coordinates of the vertices.
Cx = (a + a + d) / 3 = (2a + d) / 3

- The y-coordinate of the centroid (Cy) is the average of the y-coordinates of the vertices.
Cy = (b + c + c) / 3 = (b + 2c) / 3

Therefore, the coordinates of the centroid (Cx, Cy) are ((2a + d) / 3, (b + 2c) / 3).

**Step 2: Find the Orthocenter**
The orthocenter of a triangle is the point of intersection of its altitudes. An altitude is a line segment drawn from a vertex of the triangle perpendicular to the opposite side. To find the orthocenter, we can use the following steps:

- Calculate the slope of the line joining the points (a, b) and (a, c).
Slope = (c - b) / (a - a) = (c - b) / 0 (undefined slope)

Since the slope of the line joining (a, b) and (a, c) is undefined, it means that this line is vertical. Therefore, it is perpendicular to the line segment joining (d, c) and (a, b).

- Calculate the slope of the line joining the points (d, c) and (a, b).
Slope = (b - c) / (a - d)

The negative reciprocal of the slope of this line gives the slope of the altitude passing through the vertex (a, c).

- Calculate the equation of the altitude passing through the vertex (a, c) using the slope-intercept form.
y = mx + b, where m is the slope and b is the y-intercept.

Substituting the coordinates of the vertex (a, c) into the equation, we get:
c = (b - c) / (a - d) * a + b

Simplifying the equation, we obtain:
c = (ab - ac + ab - bd) / (a - d)

To find the y-coordinate of the orthocenter, we substitute the x-coordinate (a) of the vertex (a, c) into the equation of the altitude:
y = (b - c) / (a - d) * a + b

Therefore, the coordinates of the orthocenter are (a, (2ab - ac - bd) / (a - d)).

**Step 3: Find the Midpoint**
The midpoint of a line segment is the average of the coordinates of its endpoints. To find the midpoint of the line segment joining the centroid and orth

Given points are in right angled triangle then orthocentre (a,c) and find centroid