Ortho-centre Along with relation between all the centres

Hello maths bugs(🐞) and hivers(🐝)
I hope you are strong and stout.

Today's topic is Ortho-Centre of a triangle any of its kind.Today we gonna finish all the centres of a triangle. People get confused to differentiate them and their properties.In my previous post, I have shared lucid explanation of cir-cumcircle , in-circle , centroid. Please check them first because today I'm going to share relation between all the centres also and if you lack idea of them , reading this post may be useless for you.

What is called an Ortho-centre?

A point where meet all the three perpendiculars from the vertices to its opposite sides or extended opposite sides is called an Ortho-centre. Check it in the figure below.

Points to be considered:

1. The Ortho-centre may be inside or inside or on the side of a triangle.
2. There is no circle can be drawn whose centre is the Ortho-centre and the circle passes through the vertices unless it is not a equilateral triangle.
3. All the perpendiculars drawn from vertices to their opposite sides may not be the bisectors of respective sides unless it is an equilateral ∆.

✅✅ Ortho-centre of a right angle triangle is shown below:(Yes, it is at angle 90°)

✅✅Ortho-centre of a obtuse angle triangle is as below:(Yes, it is outside the ∆)

✅✅Ortho-centre is inside of a acute angle triangle as shown in the figure below:

RELATION BETWEEN CENTRES OF A ∆:

☑️☑️ All the centres come at the same point inside of a equilateral ∆.At that time distance between them is zero.

☑️☑️ C G O i.e cir-cum-centre , geo-centre/centroid and Ortho-centre come in a straight line irrespective of which kind of triangle it is(an equilateral ∆). In- centre may not come in the same straight line

☑️☑️ Ratio of distance between cir-cum circle & centroid and centroid & Ortho-centre is 2:1.

☑️☑️ Distance between in-centre and cir-cum-centre of a triangle(except an equilateral ∆) is square root of difference of in-radius and cir-cum-radius.Check it in the figure below:

All the figures in the post are made by me with math editor

I hope you find my post interesting.

Thank you very much for stopping by.

Next post will be soon.

All is well

Regards: @meta007